Question

The value of the limit as $x$ approaches $0$ for $\frac{\sin(5x){x}$ is}

• A. 1
• B. 5
• C. $\frac{1}{5}$
• D. 0

Hint:

Always try to express trigonometric limits in terms of $\frac{\sin x}{x}$ when $x$ approaches zero.

Answer 1

The Correct Option is B

Step 1: Use the standard limit.
A standard limit in calculus is: \[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
Step 2: Rewrite the given expression.
\[ \frac{\sin(5x)}{x} = 5 \cdot \frac{\sin(5x)}{5x} \]
Step 3: Apply the limit.
As $x \to 0$, $5x \to 0$, hence \[ \lim_{x \to 0} \frac{\sin(5x)}{5x} = 1 \]
Step 4: Final calculation.
\[ \lim_{x \to 0} \frac{\sin(5x)}{x} = 5 \times 1 = 5 \]