Math Entrance Limit

Calculus Level 3

Compute $\displaystyle \lim_{h\to 0} \frac{1}{h} \int_1^{1+2h} e^{\sqrt{x}}\sin \left(\frac{\pi {x}}{3}\right) dx$ .

0 4.708 2.718 1

1 solution

Chew-Seong Cheong
May 25, 2019

Let $f(x) = e^{\sqrt x}\sin \left(\dfrac {\pi x}3\right) \ dx$ and $\displaystyle F(x) = \int_0^x f(t) \ dt$ . Then we have:

$\begin{aligned} L & = \lim_{h \to 0} \frac 1h \int_1^{1+2h} e^{\sqrt x}\sin \left(\dfrac {\pi x}3\right) \ dx \\ & = \lim_{h \to 0} \frac {F(1+2h) - F(1)}h \\ & = 2 \color{#3D99F6} \lim_{h \to 0} \frac {F(1+2h) - F(1)}{2h} & \small \color{#3D99F6} \text{By definition of differentiation} \\ & = 2 \color{#3D99F6} f(1) \\ & = 2e\sin \frac \pi 3 \\ & \approx \boxed{4.708} \end{aligned}$

@Mukhammadsaid Jr. , you can enter as \displaystyle \lim {h\to 0} \frac 1h \int 1^{1+2h} e^{\sqrt x}\sin \left(\frac{\pi x}3 \right) dx $\displaystyle \lim_{h\to 0} \frac 1h \int_1^{1+2h} e^{\sqrt x}\sin \left(\frac{\pi x}3 \right) dx$ . Note that a lot of { } are unnecessary. \left and \right automatically adjust the brackets size. If you are using $\backslash [ \ \backslash ]$ instead of $\backslash ( \ \backslash )$ , the \displaystyle is not necessary but the formula will be centralized as follows:

$\lim_{h\to 0} \frac 1h \int_1^{1+2h} e^{\sqrt x}\sin \left(\frac{\pi x}3 \right) dx$

Chew-Seong Cheong - 2 years ago

Math Entrance Limit

1 solution

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