Question:

lim⁡x→0ex2−cosxx2=\displaystyle\lim_{x\to0}\frac{e^{x^2} -cos x}{x^{2}}=

Updated On: May 26, 2024
  • 32\frac{3}{2}
  • 12\frac{1}{2}
  • 1
  • −32-\frac{3}{2}
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The Correct Option is A

Solution and Explanation

lim⁡x→0 ex2−cos⁡xx2\displaystyle\lim _{x \rightarrow 0} \, \frac{e^{x^{2}}-\cos x}{x^{2}}

Using L'Hospital's rule,

lim⁡x→02xex2+Sinx2x \lim_{ x\rightarrow 0} \frac{2xe^{x^2} + Sinx}{2x}

lim⁡a→bex2+ lim⁡a→bSinx2x \lim_{a \rightarrow b} e^{x^2} +  \lim_{a \rightarrow b} \frac{Sinx}{2x}

=1+12=32=1+\frac1{2} = \frac3{2}

So, The correct option is A.

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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.

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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.

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