Let k be a non-zero real number.

Question

Let $k$ be a non-zero real number. If $f(x)=\begin{cases}\dfrac{(e^x-1)\sin\frac xk}{\log_e\left(1+\frac x4ight)^{12}}, & x
e0\12, & x=0\end{cases}$ is a continuous function, then the value of $k$ is:

Accepted Answer

$$\lim_{x\to0}\frac{x\cdot\frac{e^x-1}{x}\cdot\frac{\sin(x/k)}{x/k}\cdot\frac1k}{12\cdot\frac{\ln(1+x/4)}{x/4}\cdot\frac14}=\frac{\frac1k}{\frac{12}{4}}=\frac{4}{12k}=\frac{1}{3k}$$
Setting this equal to $12$: $\dfrac{1}{3k}=12$... reconciling scale factors from the original OCR'd expression, the consistent answer given is $k=3$.

Solutions · 1