solve the
Question:

$\lim _{x \rightarrow 0} \frac{x\left(e^{\left(\sqrt{1+x^{2}+x^{4}}-1\right) / x}-1\right)}{\sqrt{1+x^{2}+x^{4}}-1}$

1. (1) is equal to $\sqrt{e}$

2. (2) is equal to 1

3. (3) is equal to 0

4. (4) does not exist

Correct Option: , 2

Solution:

Let $L=\lim _{x \rightarrow 0} \frac{x\left(e^{\frac{\sqrt{1+x^{2}+x^{4}}-1}{x}}-1\right)}{\sqrt{1+x^{2}+x^{4}}-1}$

$=\lim _{x \rightarrow 0} \frac{e^{\frac{\sqrt{1+x^{2}+x^{4}}-1}{x}}-1}{\frac{\sqrt{1+x^{2}+x^{4}}-1}{x}}$

Put $\frac{\sqrt{1+x^{2}+x^{4}}-1}{x}=t$ when $x \rightarrow 0 \Rightarrow t \rightarrow 0$

$\therefore L=\lim _{t \rightarrow 0} \frac{e^{t}-1}{t}=1$