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# Limits of the form (1 power infinity) - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Limits of the form (1 power infinity) is considered one the most difficult concept.

• 195 Questions around this concept.

## Solve by difficulty

$\dpi{100} \lim_{x\rightarrow \infty }\left ( \frac{x^{2}+5x+3}{x^{2}+x+3} \right )^{\frac{1}{x}}$

If $\dpi{100} \lim_{x\rightarrow \infty }\left ( 1+\frac{a}{x}+\frac{b}{x^{2}} \right )^{2x}= e^{2},$ then the values of a and b are

Let $p= \lim_{x\rightarrow 0+}\left ( 1+\tan ^{2} \sqrt{x}\right )^{\frac{1}{2x}}$then  log p
is equal to :

## Concepts Covered - 1

Limits of the form (1 power infinity)

Limits of the form 1 (1 power infinity)

To find the limit of the form 1, we will use the following results

If $\lim_{x\rightarrow a}\;f(x)=\lim_{x\rightarrow a}\;g(x)=0$

Then,

$\lim_{x\rightarrow a}\;\left [1+f(x) \right ]^{\frac{1}{g(x)}}=e^{\lim_{\;x\rightarrow a}\frac{f(x)}{g(x)}} .$

Or

When,  $\lim_{x\rightarrow a}\;f(x)=1 and \lim_{x\rightarrow a}\;g(x)=\infty$

Then,

$\\\lim_{x\rightarrow a}\;\left [f(x) \right ]^{{g(x)}}=\lim_{x\rightarrow a}\;\left [1+(f(x)-1) \right ]^{{g(x)}}\\\\\text{}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=e^{\lim_{\;x\rightarrow a}{\left (f(x)-1 \right )}{g(x)}}$

Proof:

Let $L=\lim_{x\rightarrow a}\;\left [1+f(x) \right ]^{\frac{1}{g(x)}}$

Taking log of both sides

$log(L)=log\left( \lim_{x\rightarrow a}\;\left [1+f(x) \right ]^{\frac{1}{g(x)}}\right)$

$log(L)=\lim_{x\rightarrow a}\frac{1}{g(x)}log\left( \;1+f(x)\right)$

$log(L)=\lim_{x\rightarrow a}\frac{1}{g(x)}\left(\frac{log\left( \;1+f(x)\right)}{f(x)}\right).f(x)$

As f(x) is tending to 0, so

$log(L)=\lim_{x\rightarrow a}\frac{f(x)}{g(x)}$

$L=e^{\lim_{x\rightarrow a}\frac{f(x)}{g(x)}}$

Some particular cases

$\\\text{(a)}\;\;\;\lim_{x\rightarrow 0}\;(1+x)^{\frac{1}{x}}=e\\\\\text{(b)}\;\;\;\lim_{x\rightarrow \infty}\;\left (1+\frac{1}{x} \right )^{x}=e\\\\\text{(c)}\;\;\;\lim_{x\rightarrow 0}\;(1+cx)^{\frac{1}{x}}=e^c\\\\\text{(d)}\;\;\;\lim_{x\rightarrow \infty}\;\left (1+\frac{c}{x} \right )^{x}=e^c$

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Limits of the form (1 power infinity)

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## Books

### Reference Books

#### Limits of the form (1 power infinity)

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 2.26

Line : 32