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

JEE MainJEE Main 2020Limits and Derivatives

Solution:

$\displaystyle\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} $
$\because \displaystyle\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{2}+x^{4}}-1}{x}\left(\frac{0}{0} \text { from }\right) $
$\displaystyle\lim _{x \rightarrow 0} \frac{\left(1+x^{2}+x^{4}\right)-1}{x\left(\sqrt{1+x^{2}+x^{4}}+1\right.} $
$\displaystyle\lim _{x \rightarrow 0} \frac{x\left(1+x^{2}\right)}{\left(\sqrt{1+x^{2}+x^{4}}+1\right)}=0 $
So $\displaystyle\lim _{x \rightarrow 0} \frac{x\left(e^{\left(\frac{\sqrt{1+x^{2}+x^{4}}-1}{x}\right)}-1\right)}{\sqrt{1+x^{2}+x^{4}}-1} \quad\left(\frac{0}{0} \text { from }\right) $
$\displaystyle\lim _{x \rightarrow 0} \frac{e^{\frac{\sqrt{1+x^{2}+x^{4}-1}}{x}}-1}{\left(\frac{\sqrt{1+x^{2}+x^{4}}-1}{x}\right)}=1$