Solveeit Logo
Login

Question

Mathematics Question on limits and derivatives

lim⁡x→0(1+5x21+3x2)1x2=\displaystyle\lim_{x\to0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}} =

A

e3xe^{3x}

B

e2e^{2}

C

1e\frac{1}{e}

D

53\frac{5}{3}

Answer

e2e^{2}

Explanation

Solution

Let f(x)=(1+5x21+3x2)1x2f\left(x\right) = \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}}
∴   lim⁡x→0f(x)=lim⁡x→0(1+5x21+3x2)1x2\therefore \:\:\: \lim _{x\to 0} f\left(x\right) =\lim _{x\to 0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}}
=elim⁡x→0(1+5x21+3x2)1x2= e^{\lim _{x\to 0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}}}
=elim⁡x→0(1+5x21+3x2)1x2=e2= e^{\lim _{x\to 0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}}} = e^{2}