Question

$\underset{x\to 0}{lim}\frac{e^{x^2}-\cos x}{x^2}$ is equal to

• A. $\frac{3}{2}$
• B. $\frac{1}{2}$
• C. $\frac{2}{3}$
• D. none of these

Answer 1

Answer: (A)

$\frac{3}{2}$

$\underset{x\to 0}{lim}\frac{e^{x^2}-\cos x}{x^2}=\underset{x\to 0}{lim}\frac{e^{x^2}-1+1-\cos x}{x^2}$

$=\underset{x\to 0}{lim}(\frac{e^{x^2}-1}{x^2}+2\frac{{\sin }^2\left(x/2\right)}{x^2})$

$=\underset{x\to 0}{lim}(\frac{e^{x^2}-1}{x^2}+\frac{2}{4}{\left(\frac{{\sin }^2\left(x/2\right)}{x/2}\right)}^2)$

$=1+\frac{1}{2}=\frac{3}{2}$