$\displaystyle\lim_{x \to 0} \left(\frac{3x^{2}+2}{7x^{2}+2}\right)^{\frac {1}{x^2}}$ is equal to :
- $e$
- $\frac{1}{e^{2}}$
- $\frac{1}{e}$
- $e^{2}$
The Correct Option is B
Solution and Explanation
Required limit $=e^{\displaystyle \lim_{x \to 0}\left(\frac{3x^{2}+2}{7x^{2}+2}-1\right) \frac{1}{x^{2}} }$
$=e^{\displaystyle \lim_{x \to 0}\left(\frac{-4}{7x^{2}+2}\right)}$$=\frac{1}{e^{2}}$
The correct option is (B): \(\frac {1}{e^2}\).
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Concepts Used:
Limits And Derivatives
Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.
Limits Formula:
Derivatives of a Function:
A derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.
Properties of Derivatives:
Read More: Limits and Derivatives