Question

Evaluate :


$\displaystyle\lim_{x\to\0}\: {(1+x )}^{\frac{1}{x}}$​

Answer 1

Correct question:

$\lim_{x \to 0} (1+x)^{\dfrac{1}{x}}$

Answer:

$\boxed{e}$

Step-by-step explanation:

$\textsf{Let a\:=\:}\lim_{x \to 0}(1+x)^{\dfrac{1}x}}$

$\textsf{Taking log both sides we get :}\\\\\implies ln a=ln( \lim_{x \to 0}(1+x)^{\dfrac{1}{x}})\\\\\implies ln a= \lim_{x \to 0}(ln(1+x)^{\dfrac{1}{x}})\\\\\implies ln a= \lim_{x \to 0}(\dfrac{1}{x}ln(1+x))$

$\implies ln a= \lim_{x \to 0}(\dfrac{\dfrac{d}{dx}ln(1+x)}{\dfrac{d}{dx}(x)})\\\\\implies ln a= \lim_{x \to 0}(\dfrac{\dfrac{1}{1+x}}{1})\\\\\implies ln a= \lim_{x \to 0}(\dfrac{1}{1+x})\\\\\implies ln a= \dfrac{1}{1+0}\\\\\implies ln a=\dfrac{1}{1}\implies 1\\\\\implies a=e$

NOTE :

ln denotes natural logarithm which has base of e .

The logarithm denoted by log has base 10 .

Here L'Hospital's rule is used where we differentiate both numerator and denominator to evaluate a limit which is in indeterminate form .

The laws of logarithm and differentiation should be remembered .

Answer 2

Answer:

Step-by-step explanation:

❣Holla user❣

Here is ur ans⬇️⬇️⬇️

In last we know that:-

if the limit on right side exist so that its very easy to find the limit of other side (Through L Hospital Rule)

The derivatives of their numerator and denominator function

which is written as

f(x)=g(x)=0

so from that rule we conclude that

x => 0

Hope u like my ans^_^