Limit of a function


Introduction

In Mathematics, limit is defined as a value that a function approaches for the given point. It is always concerns about the behavior of a function at a particular point.

Meaning of \( x\to a \)

Consider a function
\(f(x)= \frac{x^2-1}{x-1}\)
We know that function is NOT defined at x=1. However, what happens to f(x) near the value x=1?

If we substitute small values for x, then we find that the value of f(x) is approximately 2 near at x=1

x<1 x>1
x 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
f(x) 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5


The closer that x gets to 1, the closer the value of the function f(x) to 2.
In such cases, we call it
f(x)=2 as x tends to 1




Intuitive Definition of Limit

Let 𝑓(π‘₯) be a function defined at all values in an open interval containing a, with the possible exception of "a" itself, and let L be a real number.
Then we say that \( \displaystyle \lim_{x \to a} f(x) =L\) if
for a given number \(\epsilon > 0\), there exists a number \(\delta > 0\)such that
|f(x)-L|<\(\epsilon\) whenever |x-a|<\(\delta\)

In this definition

  1. Given a number L, we choose Ξ΅-neighbourhood of L, Ξ΅ is positive AND can be small enough as we like such that |f(x) - L|< Ξ΅
  2. Now, we will try to find, Ξ΄-neighbourhood of a, Ξ΄ is positive AND can be small enough as satisfied such that |x - a|< Ξ΄
  3. If a small change in Ξ΅ implies a small change Ξ΄, then the limit exists at a.
  4. If a small change in Ξ΅ implies a LARGE change Ξ΄, then the limit does NOT exist at a.
How to use the applet
  1. Click on the "Show Ξ΄-neighbourhood of 'a'" check box.
  2. Given a number a, adjust Ξ΄-neighbourhood of a (drag the point a or the slider Ξ΄) , so that |x - a|< Ξ΄ , where x is any point inside the Ξ΄-neighbourhood
  3. Now, Click on the "Show Ξ΅-neighbourhood of 'L'" check box.
  4. Try to find Ξ΅-neighbourhood of L (largest distance from L) , such that |f(x) - L|< Ξ΅ , where f(x) is any point inside the Ξ΅-neighbourhood and the result is valid for all |x - a|< Ξ΄

    ΰ€Έΰ€¬ै |x - a|< Ξ΄ ΰ€•ो ΰ€²ΰ€—ी |f(x) - L|< Ξ΅  ΰ€Ήुΰ€¨े ΰ€—ΰ€°ि Ξ΅-zone ΰ€¬ाΰ€¨ΰ€‰ΰ€¨ ΰ€Έΰ€•िΰ€¨्ΰ€› ΰ€­ΰ€¨े limit exist ΰ€Ήुΰ€¨्ΰ€› ।

    Once a Ξ΅ is found, any higher Ξ΅ is always accepted.
    Once a Ξ΄ is satisfied, any smaller Ξ΄ is always accepted.
More Explanation
The intuitive definition says that
  1. determine a number Ξ΄>0
  2. take any x in the region, i.e. between a+Ξ΄ and a−Ξ΄, then this x will be closer to a, that is |x-a|<Ξ΄
  3. identify the point on the graph that our choice of x gives, then this point on the graph will lie in the intersection of the Ξ΅ region. This means that this function value f(x) will be closer to L , that is |f(x)-L|<Ξ΅
    Means
    if we take any value of x in the Ξ΄ region then the graph for those values of x will lie in the Ξ΅ region.
  4. Once a Ξ΄ is found, any smaller delta is acceptable, so there are an infinite number of possible Ξ΄'s that we can choose.
  5. the function has limit at given x



Empirical Definition of Limit

Let 𝑓(π‘₯) be a function defined at all values in an open interval containing a, with the possible exception of "a" itself, and let L be a real number.
Then we say that \( \displaystyle \lim_{x \to a} f(x) =L\) if
all of the following three conditions hold

  1. \( \displaystyle \lim_{x \to a^{-}} f(x) \) exists=LHS
  2. \( \displaystyle \lim_{x \to a^{+}} f(x) \) exists=RHS
  3. LHS=RHS
For Example
  1. In Figure (1). We see that the graph of f(x) has a hole at a. In fact, f(a) is undefined.[Limit exists at x=2]
  2. In Figure (2), f(a) is defined, but the function has a jump at a.[Limit does NOT exist at x=2]
  3. In Figure (3), f(a) is defined, but the function has a gap at a.[Limit exists at x=2]
\( f(x)=\frac{x^2-1}{x-1}\)\( \small {f(x)= \begin{cases} x+1 & \text{for } x ≤ 2 \\ x+2 & \text{for } x > 2 \end{cases}} \)\( \small { f(x)= \begin{cases} x+1 & \text{for } x \ne 2 \\ 4 & \text{for } x = 2 \end{cases}} \)



Indeterminate Form

The term "indeterminate" in mathematics refers to a situation where the value of an expression cannot be determined or uniquely identified based solely on its form or appearance.

  1. \(\frac{0}{0}\)

    In the case of the expression "\(\frac{0}{0}\)" it is called indeterminate because it doesn't provide enough information to definitively determine the value of the expression.

    For example
    \(\frac{1}{1}=1\) \(\frac{1}{0}=\infty\) \(\frac{0}{1}=0\)
    \(\frac{2}{2}=1\) \(\frac{2}{0}=\infty\) \(\frac{0}{2}=0\)
    \(\frac{3}{3}=1\) \(\frac{3}{0}=\infty\) \(\frac{0}{3}=0\)
    \(\frac{a}{a}=1\) \(\frac{a}{0}=\infty\) \(\frac{0}{a}=0\)
    \(\frac{0}{0}=1\) \(\frac{0}{0}=\infty\) \(\frac{0}{0}=0\)

    Here, \(\frac{0}{0}\) creates a situation where there is uncertainty about how the fraction \(\frac{0}{0}\) as a whole behaves. In other words, knowing that both the numerator and denominator are approaching zero doesn't immediately mean \(\frac{0}{0}\) will approach a specific finite value, approach infinity, or approach zero. The behavior of the fraction depends on the specific functions involved and how they approach zero.

  2. \(\frac{\infty}{\infty}\)
    Usually \(\frac{\infty}{number}=\infty\) and \(\frac{number}{\infty}=0\). So the top pulls the limit up to infinity and the bottom tries to pull it down to 0. So who wins?
  3. \(0.\infty\)
    Usually 0 · (number) = 0 and (number) · ∞ = ∞. So one piece tries to pull the limit down to zero, and the other tries to pull it up to ∞. Does one side win?
  4. \(\infty-\infty\)
    In general ∞ − (number) = ∞, but (number) − ∞ = −∞. So who wins?
  5. \(\infty^{0}\)
    In general ∞ raised to any positive power should be equal to ∞, ∞ raised to a negative power is 0, and anything raised to the zero should be equal to 1. So who wins?
  6. \(1^{\infty}\)
    Usually 1 raised to any power is just equal to 1. But fractions raised to the ∞ goes to zero, and numbers larger than 1 raised to the ∞ should go off to ∞. So where does \(1^{\infty}\) go?
  7. \(0^{0}\)
    In general zero raised to any positive power is just zero, but but anything raised to the zero should be equal to 1. So which is it?



Limit of algrabic function

  1. \( \displaystyle \lim_{x\to a}\frac{x^n-a^n}{x-a}=n{a^{n-1}}\)

    Solution
    We know that
    \( \frac{x^n-a^n}{x-a}=\frac{(x-a)(x^{n-1}+ax^{n-2}+a^2x^{n-3}+...+a^{n-1})}{x-a}\)
    or \( \frac{x^n-a^n}{x-a}=(x^{n-1}+ax^{n-2}+a^2x^{n-3}+...+a^{n-1})\)
    Thus, taking limit as \( x \to a\), we get
    \(\displaystyle \lim_{x\to a} \frac{x^n-a^n}{x-a}=\lim_{x\to a}(x^{n-1}+ax^{n-2}+a^2x^{n-3}+...+a^{n-1})\)
    or \(\displaystyle \lim_{x\to a} \frac{x^n-a^n}{x-a}=(a^{n-1}+a.a^{n-2}+a^2.a^{n-3}+...+a^{n-1})\)
    or \(\displaystyle \lim_{x\to a} \frac{x^n-a^n}{x-a}=n a^{n-1}\)
    This completes the proof




Limit of trigonometric function

Trigonometry is branch of mathematics that deals about Triangle. The trigonometric ratio with reference to an angle x is called trigonometric function. For example,
f(x)= sinx

In this section we learn about two very specific but important trigonometric limits, and how to use them; and other tricks to find most other limits of trigonometric functions. The first involves the sine function, and the limit is
\(\displaystyle \lim_{\theta \to 0}\frac{\sin \theta }{\theta}=1\)

Here's a graph of \(f(x)=\frac{\sin x }{x}\), showing that it has a hole at x = 0. Our task in this section will be to prove that the limit from both sides of this function is 1.

Theorems on Limit of trigonometric function

Area if triangle \(OPM=\frac{1}{2} \sin \theta \cos \theta\)

ΰ€€्ΰ€°िΰ€­ुΰ€œΰ€•ो ΰ€•्ΰ€·ेΰ€€्ΰ€°ΰ€«ΰ€² 1/2 * ΰ€†ΰ€§ाΰ€° * ΰ€‰ΰ€šाΰ€ˆ ΰ€Ήुΰ€¨े ΰ€­ΰ€ΰ€•ोΰ€²े \(\triangle OPM=\frac{1}{2} \sin \theta \cos \theta\) ΰ€Ήुΰ€¨्ΰ€› ।

Area if sector \(OPB=\frac{1}{2} \theta\)

ΰ€΅ृΰ€€ΰ€•ो ΰ€šाँΰ€¦ΰ€•्ΰ€·ेΰ€€्ΰ€°ΰ€•ो ΰ€•्ΰ€·ेΰ€€्ΰ€°ΰ€«ΰ€² 1/2 * ΰ€…ΰ€°्ΰ€§ΰ€΅्ΰ€―ाΰ€Έ 2 * ΰ€•ेΰ€¨्ΰ€¦्ΰ€°िΰ€― ΰ€•ोΰ€£ ΰ€Ήुΰ€¨े ΰ€­ΰ€ΰ€•ोΰ€²े sector \(OPB=\frac{1}{2} \theta\) ΰ€Ήुΰ€¨्ΰ€› ।

Area if triangle \(OBA=\frac{1}{2} \tan \theta\)

ΰ€€्ΰ€°िΰ€­ुΰ€œΰ€•ो ΰ€•्ΰ€·ेΰ€€्ΰ€°ΰ€«ΰ€² 1/2 * ΰ€†ΰ€§ाΰ€° * ΰ€‰ΰ€šाΰ€ˆ ΰ€Ήुΰ€¨े ΰ€­ΰ€ΰ€•ोΰ€²े \( \triangle OBA=\frac{1}{2} \tan \theta\) ΰ€Ήुΰ€¨्ΰ€› ।

In the figure above
Area of triangle OMP=\(\frac{1}{2} \sin \theta \cos \theta\)
Area of sector OAP=\(\frac{1}{2} \theta \)
Area of triangle OAB=\(\frac{1}{2} \tan \theta\)
Now
Area of triangle OMP \(\le\) Area of sector OAP \(\le\) Area of triangle OAB
or \(\frac{1}{2} \sin \theta \cos \theta \le \frac{1}{2} \theta \le \frac{1}{2} \tan \theta \)
or \(\sin \theta \cos \theta \le \theta \le \tan \theta \)
or \(\cos \theta \le \frac{\theta}{\sin \theta } \le \frac{1}{\cos \theta} \)
or \(\frac{1}{\cos \theta} \ge \frac{\sin \theta }{\theta} \ge \cos \theta \)
Taking limit as \( \theta \to 0\), we get
\( \displaystyle \lim_{\theta \to 0} \frac{1}{\cos \theta} \ge \lim_{\theta \to 0}\frac{\sin \theta }{\theta} \ge \lim_{\theta \to 0} \cos \theta \)
or \( \displaystyle \frac{1}{\cos 0} \ge \lim_{\theta \to 0}\frac{\sin \theta }{\theta} \ge \cos 0 \)
or \( \displaystyle \frac{1}{1} \ge \lim_{\theta \to 0}\frac{\sin \theta }{\theta} \ge 1 \)
or \( \displaystyle 1 \ge \lim_{\theta \to 0}\frac{\sin \theta }{\theta} \ge 1 \)
or \( \displaystyle \lim_{\theta \to 0}\frac{\sin \theta }{\theta}=1 \)
This completes the proof.




More Theorems on Limit of trigonometric function

  1. The another important limit involves the cosine function, specifically the function
    \(\displaystyle \lim_{\theta \to 0}\frac{\cos \theta -1}{\theta}=0\)

    Here's a graph of \(f(x)=\frac{\cos x-1}{x}\), showing that it has a hole at x = 0. Our task in this section will be to prove that the limit from both sides of this function is 0.

    Prove that \(\displaystyle \lim_{x \to 0}\frac{1- \cos x}{x}=0 \)
    Solution
    The limit is
    \(\displaystyle \lim_{x \to 0}\frac{1- \cos x}{x} \)
    or \(\displaystyle \lim_{x \to 0}\frac{1- \cos x}{x} \times \frac{1+ \cos x}{1+ \cos x} \)
    or \(\displaystyle \lim_{x \to 0}\frac{1- \cos^2 x}{x(1+ \cos x)} \)
    or \(\displaystyle \lim_{x \to 0}\frac{\sin^2 x}{x(1+ \cos x)} \)
    or \(\displaystyle \lim_{x \to 0}\frac{\sin x}{x} \times \lim_{x \to 0} \frac{\sin x}{1+ \cos x}\)
    or \( 1 \times \frac{0}{1+ 1}\)
    or 0
    This completes the proof

  2. \(\displaystyle \lim_{x \to 0} \sin x=0\)
  3. \(\displaystyle \lim_{x \to 0} \cos x=1\)
  4. \(\displaystyle \lim_{x \to 0}\frac{\tan x}{x}=1\)



Limit of Exponential function

A function of the form \(f (x) = a^x\) where base ‘a’ is constant (a>0) and the exponent ‘x’ is variable, is called exponential function.

For example,
\(f (x) = 2^x\)
is an exponential function.

Graph of two exponential function \(2^x, 2^{-x} \)

The great Swiss mathematician Leonhard Euler (1707-1783) has introduced the number e (e = 2.7182818284….). This value e is useful to define exponential function.
The function \(f(x)=e^x\) is called standard exponential function.
In this definition of \(f(x)=e^x\)

  1. Domain of \(f (x) = \{-\infty , \infty \}\)
  2. Range of \(f (x) = \{0, \infty \}\)
Graph of two exponential function \(e^x, e^{-x} \)
Theorem on Limit of exponential function
  1. Prove that \(\displaystyle \lim_{x \to \infty} \left( 1+\frac{1}{x} \right )^x =e\)
    Solution
    We know that
    1. \(e=1+1!+\frac{1}{2!}+\frac{1}{3!}+...\)
    2. \( (1+x)^n=1+nx+\frac{n)n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+...\)
    Thus
    \(\left( 1+\frac{1}{x} \right )^x =1+ x \frac{1}{x} +\frac{x(x-1)}{2!} \left( \frac{1}{x}\right)^2+\frac{x(x-1)(x-2)}{3!} \left( \frac{1}{x}\right)^3 +...\)
    or \( \left( 1+\frac{1}{x} \right )^x =1+1 +\frac{1(1-\frac{1}{x})}{2!} +\frac{1(1-\frac{1}{x})(1-\frac{2}{x})}{3!}+...\)
    Taking limit as \( x \to \infty \), we get
    \( \displaystyle \lim_{x \to \infty} \left( 1+\frac{1}{x} \right )^x =1+1 + \lim_{x \to \infty}\frac{1(1-\frac{1}{x})}{2!} +\lim_{x \to \infty}\frac{1(1-\frac{1}{x})(1-\frac{2}{x})}{3!}+...\)
    or \( \displaystyle \lim_{x \to \infty} \left( 1+\frac{1}{x} \right )^x =1+1 + \frac{1(1-0)}{2!} +\frac{1(1-0)(1-0)}{3!}+...\)
    or \( \displaystyle \lim_{x \to \infty} \left( 1+\frac{1}{x} \right )^x =1+1 + \frac{1}{2!} +\frac{1}{3!}+...\)
    or \( \displaystyle \lim_{x \to \infty} \left( 1+\frac{1}{x} \right )^x =e\)
    This completes the Proof.

  2. Prove that \(\displaystyle \lim_{x\to 0}\frac{e^x-1 }{x}=1 \)
    Solution
    \(\displaystyle \lim_{x\to 0}\frac{e^x-1 }{x} \)
    or \(\displaystyle \lim_{x\to 0}\frac{\left ( 1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...\right ) -1 }{x} \)
    or \(\displaystyle \lim_{x\to 0}\frac{\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+... }{x} \)
    or \(\displaystyle \lim_{x\to 0} \frac{1}{1!}+\frac{x}{2!}+\frac{x^2}{3!}+... \)
    or \(\frac{1}{1!}+\frac{0}{2!}+\frac{0^2}{3!}+... \)
    or 1
    This completes the solution.



Limit of logarithmic function

A logarithm function is an exponent of exponential function. For example,
if \( {a^x}=y\), then \(x={\log_a}y\).
In this definition
Log is the exponent, (or, exponent= Log)
if
\(3^2=9\) then \(2 = \log_3 9\)

In general, a function of the form \(f (x) = \log_e x\) called logarithmic function.
where

  • Domain of f (x) = \( (0, \infty )\)
  • Range of f (x) =\( (-\infty , \infty ) \)
Properties of logarithmic function
  1. Product property: \( \log a (x.y) = \log_ax + \log a_y \)
  2. Quotient property: \( \log_ a (x/y) = \log_ax - \log_ay \)
  3. Power property: \( \log _ ax^nn = n \log _ax \)
  4. \( \log_ a a = 1, \log_ a 1 = 0 \)
  5. \( \log_ a m = \log_ a b \times \log_ b m \)
Graph of exponential and logarithem function

Log is the reflection of exponential function about y=x line, which is shown in a graph given below

Theorems on Limit of logarithmic function
  1. \(\displaystyle \lim_{x\to 0}\frac{\log_e(1+x)}{x}=1 \)
  2. \(\displaystyle \lim_{x\to 0}\frac{\log_e(1-x)}{-x}=1 \)
  3. \(\displaystyle \lim_{x\to 0}\frac{a^x-1}{x}=\log a\)
    We know that
    \( \frac{d}{dx} a^x= \frac{d}{dx} e^{\log (a^x)} \)
    or \( \frac{d}{dx} a^x= \frac{d}{dx} e^{x \log a} \)
    or \( \frac{d}{dx} a^x= \log a .e^{x \log a} \)
    or \( \frac{d}{dx} a^x= \log a .e^{\log a^x} \)
    or \( \frac{d}{dx} a^x= \log a. a^x \)
    Thus, the limit is
    \(\displaystyle \lim_{x\to 0}\frac{a^x-1}{x}\)
    or \(\displaystyle \lim_{x\to 0}\frac{ \frac{d}{dx}(a^x-1)}{\frac{d}{dx} (x)} \)
    or \(\displaystyle \lim_{x\to 0}\frac{ \log a . a^x}{1} \)
    or \(\log a .a^0 \)
    or \(\log a \)
    This completes the proof



Limit (BCB-Revised Edition 2020, Exercise 2, Page 379)

Evaluate the following.
  1. \( \displaystyle \lim_{x \to 0} \frac{\sin ax}{x} \)

    Solution πŸ‘‰ Click Here

  2. \( \displaystyle \lim_{x \to 0} \frac{\tan bx}{x} \)

    Solution πŸ‘‰ Click Here

  3. \( \displaystyle \lim_{x \to 0} \frac{\sin mx}{\sin nx} \)

    Solution πŸ‘‰ Click Here

  4. \( \displaystyle \lim_{x \to 0} \frac{\tan ax}{\tan bx} \)

    Solution πŸ‘‰ Click Here

  5. \( \displaystyle \lim_{x \to 0} \frac{\sin px}{\tan qx} \)

    Solution πŸ‘‰ Click Here

  6. \( \displaystyle \lim_{x \to a} \frac{\sin (x-a)}{x^2-a^2} \)

    Solution πŸ‘‰ Click Here

  7. \( \displaystyle \lim_{x \to p} \frac{x^2-p^2}{\tan (x-p)} \)

    Solution πŸ‘‰ Click Here

  8. \( \displaystyle \lim_{x \to 0} \frac{\sin ax. \cos bx}{\sin cx} \)

    Solution πŸ‘‰ Click Here


  9. \( \displaystyle \lim_{x \to 0} \frac{1-\cos x}{x^2} \)

    Solution πŸ‘‰ Click Here

  10. \( \displaystyle \lim_{x \to 0} \frac{1-\cos 6x}{x^2} \)

    Solution πŸ‘‰ Click Here

  11. \( \displaystyle \lim_{x \to 0} \frac{1-\cos 9x}{x^2} \)

    Solution πŸ‘‰ Click Here

  12. \( \displaystyle \lim_{x \to 0} \frac{\cos ax-\cos bx}{x^2} \)

    Solution πŸ‘‰ Click Here

  13. \( \displaystyle \lim_{x \to 0} \frac{\sin ax-\sin bx}{x} \)

    Solution πŸ‘‰ Click Here

  14. \( \displaystyle \lim_{x \to 0} \frac{1-\cos px}{1-\cos qx} \)

    Solution πŸ‘‰ Click Here


  15. \( \displaystyle \lim_{x \to 0} \frac{\tan x-\sin x}{x^3} \)

    Solution πŸ‘‰ Click Here

  16. \( \displaystyle \lim_{x \to 0} \frac{\tan 2x-\sin 2x}{x^3} \)

    Solution πŸ‘‰ Click Here

  17. \( \displaystyle \lim_{x \to \frac{\pi}{2}} (\sec x -\tan x) \)

    Solution πŸ‘‰ Click Here

  18. \( \displaystyle \lim_{x \to \frac{\pi}{4}} \frac{\sec ^2 x-2}{\tan x-1}\)

    Solution πŸ‘‰ Click Here

  19. \( \displaystyle \lim_{x \to \frac{\pi}{4}} \frac{2- \csc ^2 x}{1 -\cot x}\)

    Solution πŸ‘‰ Click Here

  20. \( \displaystyle \lim_{x \to y} \frac{\tan x -\tan y}{x-y}\)

    Solution πŸ‘‰ Click Here

  21. \( \displaystyle \lim_{x \to y} \frac{\sin x -\sin y}{x-y}\)

    Solution πŸ‘‰ Click Here

  22. \( \displaystyle \lim_{x \to y} \frac{\cos x -\cos y}{x-y}\)

    Solution πŸ‘‰ Click Here

  23. \( \displaystyle \lim_{x \to \theta} \frac{x \cot \theta -\theta \cot x}{x -\theta}\)

    Solution πŸ‘‰ Click Here

  24. \( \displaystyle \lim_{x \to \theta} \frac{x \cos \theta -\theta \cos x}{x -\theta}\)

    Solution πŸ‘‰ Click Here

  25. \( \displaystyle \lim_{x \to 1} \frac{1+ \cos \pi x}{\tan ^2 \pi x}\)

    Solution πŸ‘‰ Click Here

  26. \( \displaystyle \lim_{x \to \theta} \frac{x \tan \theta -\theta \tan x}{x -\theta}\)

    Solution πŸ‘‰ Click Here

  27. \( \displaystyle \lim_{\theta \to \frac{\pi}{4}} \frac{\cos \theta -\sin \theta}{\theta -\frac{\pi}{4}}\)

    Solution πŸ‘‰ Click Here

  28. \( \displaystyle \lim_{x \to c} \frac{\sqrt{x}-\sqrt{c}}{\sin x-\sin c}\)

    Solution πŸ‘‰ Click Here


  29. Find the limits of
    1. \( \displaystyle \lim_{x \to 0} \frac{e^{6x}-1}{x}\)

      Solution πŸ‘‰ Click Here

    2. \( \displaystyle \lim_{x \to 0} \frac{e^{2x}-1}{x. 2^{x+1}}\)

      Solution πŸ‘‰ Click Here

    3. \( \displaystyle \lim_{x \to 0} \frac{e^{ax}-e^{bx}}{x}\)

      Solution πŸ‘‰ Click Here


    4. \( \displaystyle \lim_{x \to 0} \frac{a^x+b^x-2}{x}\)

      Solution πŸ‘‰ Click Here

  30. Evaluate the limits of
    1. \( \displaystyle \lim_{x \to 2} \frac{x-2}{\log (x-1)}\)

      Solution πŸ‘‰ Click Here

    2. \( \displaystyle \lim_{x \to \frac{\pi}{2}} \frac{\cos x}{\log \left ( x- \frac{\pi}{2} +1\right )}\)

      Solution πŸ‘‰ Click Here




Additional Question (Limit)[Page 392]

  1. Prove that \( \displaystyle \lim_{x \to \frac{2}{3}} \frac{2}{2-3x}\) does NOT exist.

    Solution πŸ‘‰ Click Here

  2. Do the following function define for the value x=1?
    1. \( f(x)=\frac{x-1}{x+2}\)

      Solution πŸ‘‰ Click Here

    2. \( f(x)=\frac{x^3+1}{x-1}\)

      Solution πŸ‘‰ Click Here

  3. What do you mean by the left hand limit and right hand limit of a function? What is the condition for the limit of a function to exist at a point?
    Prove that \(\displaystyle \lim_{x \to 0}|x|=0\) but \(\displaystyle \lim_{x \to 0} \frac{|x|}{x} \) does not exist.

    Solution πŸ‘‰ Click Here

  4. Distinguish between limit and value of a function.
    It is given that \(f(x)=\frac{ax+b}{x+1},\displaystyle \lim_{x \to 0} f(x)=2\) and \( \displaystyle \lim_{x \to \infty} f(x)=1\). Prove that \(f(-2)=0\)

    Solution πŸ‘‰ Click Here

  5. Define limit of a function at a point. It is given that \(f(x)=\frac{x+6}{cx-d},\displaystyle \lim_{x \to 0} f(x)=-6 \) and \( \displaystyle \lim_{x \to \infty} f(x)=\frac{1}{3}\).
    Prove that \(f(13)=\frac{1}{2}\)

    Solution πŸ‘‰ Click Here

  6. What do you mean by an indeterminate form? State their different forms. Evaluate the following limit \(\displaystyle \lim_{x \to \infty} \sqrt{x} (\sqrt{x}-\sqrt{x-a})\)

    Solution πŸ‘‰ Click Here

  7. Let \(f:R \to R \) be defined by \(f(x)=\begin{cases} x & \text{if x is an integer} \\ 0 & \text{if x is not an integer} \\ \end{cases}\)
    Find \( \displaystyle \lim_{x \to 1} f(x)\). Is it same as \(f(1)\)

    Solution πŸ‘‰ Click Here

  8. Prove that
    1. \(\displaystyle \lim_{x \to 3} \left ( \frac{1}{x-3}-\frac{9}{x^3-3x^2}\right ) =\frac{2}{3}\)

      Solution πŸ‘‰ Click Here

    2. \(\displaystyle \lim_{x \to 3} \left ( \frac{x^2+9}{x^2-9}-\frac{3}{x-3}\right )=\frac{1}{2}\)

      Solution πŸ‘‰ Click Here

  9. Evaluate
    1. \(\displaystyle \lim_{x \to 2} \frac{x^{-3}-2^{-3}}{x-2}\)

      Solution πŸ‘‰ Click Here

    2. \(\displaystyle \lim_{x \to \infty} \frac{(2x-1)^6(3x-1)^4}{(2x+1)^{10}}\)

      Solution πŸ‘‰ Click Here

    3. \(\displaystyle \lim_{x \to 0} \frac{(1+x)^6-1}{(1+x)^2-1}\)

      Solution πŸ‘‰ Click Here

    4. \(\displaystyle \lim_{x \to a} \frac{(x+2)^{\frac{5}{2}} -(a+2)^{\frac{5}{2}} }{x-a}\)

      Solution πŸ‘‰ Click Here

  10. If \( \displaystyle \lim_{x \to a} \frac{x^3-a^3}{x-a}=27\), Find all possible values of a.

    Solution πŸ‘‰ Click Here

  11. Find the limiting values of
    1. \( \displaystyle \lim_{x \to 0} \frac{\sin x^0}{x}\)

      Solution πŸ‘‰ Click Here

    2. \( \displaystyle \lim_{x \to 0} \frac{1-\cos 4 x}{1-\cos 6x}\)

      Solution πŸ‘‰ Click Here

    3. \( \displaystyle \lim_{x \to \frac{\pi}{2}} \frac{\cos x}{\frac{\pi}{2}-x}\)

      Solution πŸ‘‰ Click Here

    4. \( \displaystyle \lim_{x \to 0} \frac{\tan 2 x-x}{3x-\sin x}\)

      Solution πŸ‘‰ Click Here

    5. \( \displaystyle \lim_{x \to \pi} \frac{1-\sin (\frac{x}{2})}{(\pi-x)^2}\)

      Solution πŸ‘‰ Click Here

    6. \( \displaystyle \lim_{x \to \frac{\pi}{2}} \frac{1+\cos 2x}{(\pi-2x)^2}\)

      Solution πŸ‘‰ Click Here

    7. \( \displaystyle \lim_{x \to 0} \sin (\frac{1}{x})\)

      Solution πŸ‘‰ Click Here

    8. \( \displaystyle \lim_{x \to 0} x \sin (\frac{1}{x})\)

      Solution πŸ‘‰ Click Here

    9. \( \displaystyle \lim_{x \to a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}\)

      Solution πŸ‘‰ Click Here

    10. \( \displaystyle \lim_{x \to a} (a-x)\tan (\frac{\pi x}{2a})\)

      Solution πŸ‘‰ Click Here

    11. \( \displaystyle \lim_{y \to 0} \frac{(x+y)\sec (x+y)-x \sec x}{y}\)

      Solution πŸ‘‰ Click Here


    1. A function is defined as \(f(x)=\begin{cases} 3x^2+2 & \text { if } x<1 \\ 2x+3 & \text { if } x \ge 1 \end{cases}\).
      Find \(\displaystyle \lim_{x \to 1}f(x) \)

      Solution πŸ‘‰ Click Here

    2. A function is defined as \(f(x)=\begin{cases} 3+2x & \text { if } -\frac{3}{2} \le x < 0\\ 3-2x & \text { if } 0 \le x < \frac{3}{2} \\ -3-2x & \text { if } x \ge \frac{3}{2} \end{cases}\).
      Find \(\displaystyle \lim_{x \to 0}f(x) \) and \(\displaystyle \lim_{x \to \frac{3}{2}}f(x) \) if they exist.

      Solution πŸ‘‰ Click Here


    1. \(\displaystyle \lim_{x \to 0} \frac{e^{px}-1}{e^{qx}-1} \)

      Solution πŸ‘‰ Click Here

    2. \(\displaystyle \lim_{x \to 0} \frac{e^x-e^{-x}-x}{x} \)

      Solution πŸ‘‰ Click Here

    3. \(\displaystyle \lim_{x \to 0} \frac{a^x-1}{b^x-1} \)

      Solution πŸ‘‰ Click Here


    1. \(\displaystyle \lim_{x \to 0} \frac{2^x-1}{\sin x} \)

      Solution πŸ‘‰ Click Here

    2. \(\displaystyle \lim_{x \to 0} \frac{e^{\sin x} -\sin x -1}{x} \)

      Solution πŸ‘‰ Click Here

    3. \(\displaystyle \lim_{x \to \frac{\pi}{2}} \frac{e^{\cos x} -1}{\frac{\pi}{2}-x} \)

      Solution πŸ‘‰ Click Here

    4. \(\displaystyle \lim_{x \to e} \frac{\log x-1}{x-e} \)

      Solution πŸ‘‰ Click Here

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