It's $x^2$ in the Denominator

Calculus Level 2

$\large \lim_{x \to 0} \dfrac{e^x + e^{-x} - 2}{x^2} = \, ?$

0 1 2

\infty

2 solutions

Guillermo Templado
Mar 8, 2016

1.- $\lim_{x \to 0} \frac{e^x + e^{-x} - 2}{x^2} = \lim_{x \to 0} \frac{1 + x + \frac{x^2}{2!} + 1 - x + \frac{(-x)^2}{2!} - 2}{x^2} = 1$ I have used $e^x$ ~ $1 + x + \frac{x^2}{2!}$ when $x \to 0$ applying McLaurin series

2.- $\lim_{x \to 0} \frac{e^x + e^{-x} - 2}{x^2} = \lim_{x \to 0} \frac{2 \cos ix - 2}{x^2} = \lim_{x \to 0} \frac{2(\cos ix -1)}{x^2} = \lim_{x \to 0} \frac{2(\frac{-(ix)^2}{2!})}{x^2} = 1$ I have used $\cos x$ ~ $1 - \frac{x^2}{2!}$ when $x \to 0$

3.- Use L'Hopital's Rule twice

I have got another :), write it as:- $\lim_{x\to 0}\left(\dfrac{(e^x-1)}{x}\right)^2\times\dfrac {1}{e^x}=1$ (Using $\lim_{x\to 0} \dfrac{e^x-1}{x}=1$ ).

Rishabh Jain - 5 years, 3 months ago

Approach 4: Use $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$ .

Nihar Mahajan - 5 years, 3 months ago

Another approach: Note that the original form allows us to use L'Hopital's Rule. Thus, $\lim_{x \to 0} \frac{e^x + e^{-x} - 2}{x^2}=\lim_{x \to 0} \frac{\frac{\text{d}(e^x + e^{-x} - 2)}{\text{d}x}}{\frac{\text{d}x^2}{\text{d}x}}=\lim_{x \to 0} \frac{e^x - e^{-x} }{2x}=\lim_{x \to 0} \frac{e^x + e^{-x}}{2}=1$

A Former Brilliant Member - 5 years, 3 months ago

It is incorrect to write $\dfrac{d \ \text{expression}}{dx}$ . You should, instead, write $\dfrac{d}{dx} (\text{expression})$ . Also, $\dfrac{d}{dx} (e^x + e^{-x} - 2) = e^x - e^{-x} \neq e^x - e^{-x} - 2$ .

This is equivalent to Guillermo Templado's third method.

Jake Lai - 5 years, 3 months ago

To the person who posted the question: please change the current $n \rightarrow 0$ to $x \rightarrow 0$ .

Shaun Leong - 5 years, 3 months ago

Thanks , I have edited it. (Its surprising I didn't notice it O.o)

Nihar Mahajan - 5 years, 3 months ago

. .
Mar 15, 2021

$\displaystyle \lim _ { x \to 0 } \frac { e ^ { x } + e ^ { -x } - 2 } { x ^ { 2 } } = 1$ .

It's x 2 x^2 x 2 in the Denominator

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It's $x^2$ in the Denominator