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Indeterminate forms of Limits

Examples with detailed solutions and exercises that solves limits questions related to indeterminate forms such as :

Theorem

A second version of L'Hopital's rule allows us to replace the limit problem

with another simpler problem to solve.
Second Version of l'Hopital theorem

Examples with Solutions

Example 1

Find the limit
Find Limit of ln(x)/x

Solution to Example 1:
Since

and

we have the indeterminate form

Find Limit of ln(x)/x , Step 3
The above L'Hopital's rule can be used to evaluate the given limit question as follows
Find Limit of ln(x)/x , Step 4
Evaluate the derivatives in the numerator and the denominator
Find Limit of ln(x)/x , Step 5
Evaluate the limits in the numerator and denominator

We now evaluate the given limit
Find Limit of ln(x)/x , Step 7

Example 2

Find

$\lim_{x\to \infty} x e^{-x}$

Solution to Example 2:
Note that

$\lim_{x\to \infty} x = \infty$
and

$\lim_{x\to \infty} e^{-x} = 0$
This is the indeterminate form

$\infty \cdot 0$ . The idea is to convert it into to the indeterminate form

$\dfrac{\infty}{\infty}$ and use L'Hopital's theorem. Note that

$\lim_{x\to \infty} x e^{-x} = \lim_{x\to \infty} \dfrac{ x}{ e^x} = \dfrac{ \infty }{ \infty }$
We apply the above L'Hopital's theorem

$= \lim_{x\to \infty} \dfrac{ (x)'}{ (e^x)'} = \lim_{x\to \infty} \dfrac{ 1 }{ e^x} = 0$

Example 3

Find

$\lim_{x\to \infty} ( 1 + 1/x)^x$

Solution to Example 3:
Note that $\lim_{x\to \infty} (1 + 1/x) = 1$ and the above limit is given by
$\lim_{x\to \infty} ( 1 + 1/x)^x = 1^{\infty}$
which is of the Indeterminate form $1^{\infty}$ . If we let $t = 1 / x$ the above limit may written as
$\lim_{x\to \infty} ( 1 + 1/x)^x = \lim_{t\to 0} ( 1 + t)^{1/t}$ , note that as $x\to \infty$ , $t\to 0$
Let $y = ( 1 + t)^{1/t}$ and find the limit of $\ln y$ as t approaches 0
$\ln y = \ln ( 1 + t)^{1/t} =(1 / t) \ln (1 + t)$
The advantage of using $\ln y$ is that
$\lim_{t\to 0} \ln y = \lim_{t\to 0} \dfrac {\ln (1 + t)}{t} = \dfrac{0}{0}$
and the limit has the indeterminate form 0 / 0 and the first L'Hopital's rule can be applied
$\lim_{t\to 0} \ln y = \lim_{t\to 0} \dfrac {ln (1 + t)}{t}$
$= \lim_{t\to 0} \dfrac{(ln (1 + t))'}{(t)'} = \lim_{t\to 0} \dfrac{1/(1+t)}{1} = 1$
Since the limit of ln y = 1 the limit of y is e¹ = e, hence
$\lim_{x\to \infty} ( 1 + 1/x)^x = e$

Example 4

Find the limit

$\lim_{x\to 0^+} ( 1 / x - 1 / \sin x )$
Solution to Example 4:
Note that

$\lim_{x\to 0^+} ( 1 / x ) = + \infty$
and

$\lim_{x\to 0^+} ( 1 / \sin x ) = + \infty$
This limit has the indeterminate form

$\infty - \infty$ and has to be converted to another form by combining

$1 / x - 1 / sin x$

$\lim_{x\to 0^+} ( 1 / x - 1 / \sin x ) = \lim_{x\to 0^+} \dfrac{\sin x - x}{x \sin x} = \dfrac{0}{0}$
We now have the indeterminate form 0 / 0 and we can use the L'Hopital's theorem.

$\lim_{x\to 0^+} \dfrac{\sin x - x}{x \sin x} = \lim_{x\to 0^+} \dfrac{(\sin x - x)'}{(x \sin x)'} = \lim_{x\to 0^+} \dfrac{\cos x - 1}{ \sin x + x \cos x} = \dfrac{0}{0}$
We have again the indeterminate form 0 / 0 and use the L'Hopital's theorem one more time.

$= \lim_{x\to 0^+} \dfrac{(\cos x - 1)'}{ (\sin x + x \cos x)'}$

$= \lim_{x\to 0^+} \dfrac{-\sin x}{ \cos x + \cos x - x \sin x } = \dfrac{0}{2} = 0$

Example 5

Find the limit

$\lim_{x\to 0^+} x^x$
Solution to Example 5:
We have the indeterminate form 0⁰. Let y = x^x and ln y = ln (x^x) = x ln x. Let us now find the limit of ln y

$\lim_{x\to 0^+} \ln y = \lim_{x\to 0^+} x \ln x = 0 \cdot \infty$
The above limit has the indeterminate form

$0 \cdot \infty$ . We have convert it as follows

$\lim_{x\to 0^+} x \ln x = \lim_{x\to 0^+} \dfrac{\ln x}{1/x} = \dfrac{\infty}{\infty}$
It now has the indeterminate form

$\dfrac{\infty}{\infty}$ and we can use the L'Hopital's theorem

$\lim_{x\to 0^+} \dfrac{\ln x}{1/x} = \lim_{x\to 0^+} \dfrac{(\ln x)'}{(1/x)'}$

$= \lim_{x\to 0^+} \dfrac{1/x}{-1/x^2}$

$= \lim_{x\to 0^+} - x = 0$
The limit of ln y = 0 and the limit of y = x^x is equal to

$e^0 = 1$

Exercises

Find the limits
1.

$\lim_{x\to \infty} (\ln x)^{1/x}$
2.

$\lim_{x\to \infty} (\ln x - \ln (1 + x))$
3.

$\lim_{x\to \infty} \dfrac{x}{e^x}$
4.

$\lim_{x\to 0^+} x^{\sin x}$

Solutions to Above Exercises

1. 1
2. 0
3. 0
4. 1