question 10 - calculator allowed. lim x->0 x/sinx is (a) -1 (b) 0 (c) 1/2 (d) 1 (e) 2
question 10 - calculator allowed. lim x->0 x/sinx is (a) -1 (b) 0 (c) 1/2 (d) 1 (e) 2
Answer
Explanation:
Step1: Recall the well - known limit
The limit $\lim_{x\rightarrow0}\frac{\sin x}{x}=1$ is a fundamental result in calculus. We can rewrite the given limit $\lim_{x\rightarrow0}\frac{x}{\sin x}$ as $\lim_{x\rightarrow0}\frac{1}{\frac{\sin x}{x}}$.
Step2: Use the limit property
If $\lim_{x\rightarrow a}f(x) = L\neq0$, then $\lim_{x\rightarrow a}\frac{1}{f(x)}=\frac{1}{L}$. Here, since $\lim_{x\rightarrow0}\frac{\sin x}{x}=1$, we have $\lim_{x\rightarrow0}\frac{1}{\frac{\sin x}{x}}=\frac{1}{1}$.
Answer:
D. 1