Question

$\lim_{x\to0}\frac{x^{100}\sin 7x}{(\sin x)^{101}}$ is equal to

• A. 7
• B. $\frac{1}{7}$
• C. 14
• D. 1
• E. 0

Hint:

Calculus Tip: When finding limits of the form $\lim_{x\to0} \frac{\sin(ax)}{\sin(bx)}$, the $x$'s cancel out and the limit instantly simplifies to the ratio of the coefficients: $\frac{a}{b}$.

Answer 1

The Correct Option is A

Concept:
To evaluate trigonometric limits approaching zero, we use the standard limit property $\lim_{x\to0} \frac{\sin(ax)}{ax} = 1$. We can manipulate the given expression by splitting the fraction and grouping the terms to match this standard form.

Step 1: Rewrite the limit expression.
Separate the power terms and the sine terms to make grouping easier: $$L = \lim_{x\rightarrow0} \frac{x^{100}}{(\sin x)^{100}} \cdot \frac{\sin 7x}{\sin x}$$

Step 2: Group the terms into standard limit forms.
Combine the terms with the power of 100, and introduce $x/x$ to balance the remaining sine terms: $$L = \lim_{x\rightarrow0} \left(\frac{x}{\sin x}\right)^{100} \cdot \left(\frac{\sin 7x}{x}\right) \cdot \left(\frac{x}{\sin x}\right)$$

Step 3: Evaluate the standard limit components.
We know that $\lim_{x\to0} \frac{x}{\sin x} = 1$. Substitute this into the limit: $$L = (1)^{100} \cdot \lim_{x\rightarrow0} \left(\frac{\sin 7x}{x}\right) \cdot (1)$$

Step 4: Adjust the remaining limit for evaluation.
To use the property $\frac{\sin(ax)}{ax} = 1$, multiply the numerator and denominator by 7: $$L = 1 \cdot \lim_{x\rightarrow0} \left(\frac{\sin 7x}{7x} \cdot 7\right)$$

Step 5: Calculate the final limit.
Evaluate the final limit, knowing that $\frac{\sin 7x}{7x} \to 1$: $$L = 1 \cdot (1 \cdot 7)$$ $$L = 7$$ Hence the correct answer is (A) 7.