Limits
Limit of exponential form — finding integer k
MJAT_TS7_P2
Grade 12
Question:
Let $k\in\mathbb{Z}$. If $\displaystyle\lim_{x\to 0^+}(\sin(kx)+\cos x+x)^{2/x}=e^6$, then the value of $k$ is:
Step-by-Step Solution
Key Concept: $\lim_{x\to 0^+}(\sin kx+\cos x+x)^{2/x}=e^{\lim(2/x)\ln(\sin kx+\cos x+x)}$. As $x\to 0^+$: $\sin kx+\cos x+x\approx 1+(k+1)x$. So $\ln(...)\approx(k+1)x$ and $\lim(2/x)\cdot(k+1)x=2(k+1)=6\Rightarrow k+1=3\Rightarrow k=2$.
$k=\mathbf{2}$. Answer: **B**.
Correct Answer: B