Limit of a Definite Integral

Calculus Level 2

$\large \lim_{m \to 0} \frac{1}{m} \int_{\sqrt{\frac{\pi}{2}}}^{\sqrt{\frac{\pi}{2}}+m} \cos (x^2) \ dx = ?$

The answer is 0.

1 solution

Chew-Seong Cheong
Jan 27, 2019

$\begin{aligned} L & = \lim_{m \to 0} \frac 1m \int_{\sqrt{\frac \pi 2}}^{\sqrt{\frac \pi 2}+m} \cos (x^2) \ dx & \small \color{#3D99F6} \text{A 0/0 case, L'Hôpital's rule applies.} \\ & = \lim_{m \to 0} \frac {\cos \left(\left(\sqrt{\frac \pi 2}+m\right)^2\right)}1 & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }m. \\ & = \frac {\cos \left(\frac \pi 2\right)}1 = \boxed 0 \end{aligned}$

Reference: L'Hôpital's rule

@Chew-Seong Cheong please explain how it is a 0/0 case?

Raja the king - 2 years, 4 months ago

Note that $\displaystyle \lim_{m \to 0} \int_a^{a+m} f(x) \ dx = 0$ and $\displaystyle \lim_{m \to 0} m = 0$ . I missed out the $\dfrac 1m$ earlier.

Chew-Seong Cheong - 2 years, 4 months ago

Limit of a Definite Integral

The answer is 0.

1 solution

0 pending reports