#2015_29 Continuity at a point

Calculus Level 3

$\large{ f(x) = \begin{cases} \dfrac{1-\sin^3(x)}{3\cos^2(x)} & \text{for } x<\dfrac\pi2 \\ a & \text{for } x=\dfrac\pi2 \\ \dfrac{b(1-\sin(x))}{(\pi-2x)^2} & \text{for } x>\dfrac\pi2\\ \end{cases}}$

Find the sum of values of $a$ and $b$ such that the function $f(x)$ above is continuous at $x=\dfrac { \pi }{ 2 }$ .

The answer is 4.5.

1 solution

Ayon Ghosh
Dec 1, 2017

For $f(x)$ to be continuous at $x = \pi/2$ we must have $\lim_{x\rightarrow \pi/2^-} f(x) = \lim_{x\rightarrow \pi/2^+} f(x) = f(\pi/2) = a$

To evaluate the first 2 limits we must resort to L Hospital 's since they are of $0/0$ form.These yield

$\lim_{x\rightarrow \pi/2^-} f(x) = 1/2 ; \lim_{x\rightarrow \pi/2^-} f(x) = b/8$

Solving for $a$ and $b$ we get $a = 1/2$ and $b = 4$ ;

$\boxed{a + b = 4.5}$ .

Hey, Dec 2 is my birthday :P . Lol, You wrote the solution that day :D

Md Zuhair - 2 years, 11 months ago

Mine is december 5!

Swapnil Das - 2 years, 11 months ago

#2015_29 Continuity at a point

The answer is 4.5.

1 solution

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