$\displaystyle\lim_{x\to0}\frac{e^{x^2} -cos x}{x^{2}}=$
- $\frac{3}{2}$
- $\frac{1}{2}$
- 1
- $-\frac{3}{2}$
The Correct Option is A
Solution and Explanation
\(\displaystyle\lim _{x \rightarrow 0} \, \frac{e^{x^{2}}-\cos x}{x^{2}}\)
Using L'Hospital's rule,
\( \lim_{ x\rightarrow 0} \frac{2xe^{x^2} + Sinx}{2x}\)
\( \lim_{a \rightarrow b} e^{x^2} + \lim_{a \rightarrow b} \frac{Sinx}{2x}\)
\(=1+\frac1{2} = \frac3{2}\)
So, The correct option is A.
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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