Question

Evaluate:$\underset{x\to 0}{\lim }\frac{e^{x^2}-\cos x}{x^2}$

• A. $0$
• B. $\frac{1}{2}$
• C. $1$
• D. $\frac{3}{2}$

Answer 1

The correct option is D

$\frac{3}{2}$

Explanation for the correct answer:

Simplifying the equation to determinate form and applying the limits:

$$\begin{gathered} \underset{x\to 0}{\Rightarrow \lim }\frac{e^{x^2}-\cos x}{x^2}\left[\text{This is in}\frac{0}{0}\text{form}\right] \\ \Rightarrow \underset{x\to 0}{\lim }\frac{e^{x^2}\left(2x\right)+\sin x}{2x}\left[\text{Differentiating}\right]\left[\text{This is in}\frac{0}{0}\text{form}\right] \\ \Rightarrow \underset{x\to 0}{\lim }\frac{e^{x^2}\left(2\right)+2x\left(e^{x^2}\left(2x\right)\right)+\cos x}{2}\left[\text{Differentiating}\right] \end{gathered}$$

Applying the limits

$$\begin{gathered} \Rightarrow \frac{e^0\left(2\right)+2\left(0\right)\left(e^0\left(0\right)\right)+\cos 0}{2} \\ \Rightarrow \frac{1\left(2\right)+0+1}{2} \\ \Rightarrow \frac{2+1}{2} \\ \Rightarrow \frac{3}{2} \end{gathered}$$

Thus, $\underset{x\to 0}{\lim }\frac{e^{x^2}-\cos x}{x^2}=\frac{3}{2}$

Therefore, the correct answer is option (D).