Let [t] denote the greatest integer less than or equal to t.

Question

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function
$$f(x)=\begin{cases}b^2\sin\left(\dfrac{\pi}{2}\left[\dfrac{\pi}{2}(\cos x+\sin x)\cos x\right]\right), & x<0\[2mm] \dfrac{\sin x-\dfrac12\sin2x}{x^3}, & x>0\[2mm] a, & x=0\end{cases}$$
is continuous at $x=0$, then $a^2+b^2$ is equal to:

Accepted Answer

Left-hand limit ($x\to0^-$): as $x\to0^-$, $\cos^2x\to1^-$ and $\sin2x\to0^-$, so the greatest-integer argument $\dfrac{\pi}{2}\left(\cos^2x+\dfrac{\sin2x}{2}\right)\to\dfrac{\pi}{2}\approx1.57^-$, whose floor is $1$. Hence
$$\text{LHL}=b^2\sin\left(\frac{\pi}{2}\cdot1\right)=b^2$$

Right-hand limit ($x\to0^+$):
$$\lim_{x\to0^+}\frac{\sin x-\frac12\sin2x}{x^3}=\lim_{x\to0^+}\frac{\sin x(1-\cos x)}{x^3}=\lim_{x\to0^+}\frac{\sin x}{x}\cdot\frac{1-\cos x}{x^2}=1\cdot\frac12=\frac12$$

By continuity, $a=b^2=\dfrac12$, so
$$a^2+b^2=a^2+a=\frac14+\frac12=\frac34$$

Solutions · 1