Question

$\lim_{x \to \infty} x \tan \left( \frac{1}{x} \right) = $ _____

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 3 \)

Hint:

When dealing with limits involving trigonometric functions as $x \to \infty$ or $x \to 0$, try using standard limits like $\lim_{t \to 0} \frac{\sin(t)}{t} = 1$ by appropriate substitution.

Answer 1

The Correct Option is B

We need to evaluate the limit $\lim_{x \to \infty} x \tan \left( \frac{1}{x} \right)$. Let $t = \frac{1}{x}$. As $x \to \infty$, $t \to 0$. So the limit becomes: $$\lim_{t \to 0} \frac{1}{t} \tan(t) = \lim_{t \to 0} \frac{\tan(t)}{t}$$ We know that $\lim_{t \to 0} \frac{\sin(t)}{t} = 1$ and $\lim_{t \to 0} \cos(t) = 1$. We can rewrite the expression as: $$\lim_{t \to 0} \frac{\tan(t)}{t} = \lim_{t \to 0} \frac{\sin(t)}{t \cos(t)} = \lim_{t \to 0} \left( \frac{\sin(t)}{t} \cdot \frac{1}{\cos(t)} \right)$$ Using the limit properties: $$= \left( \lim_{t \to 0} \frac{\sin(t)}{t} \right) \cdot \left( \lim_{t \to 0} \frac{1}{\cos(t)} \right)$$ $$= (1) \cdot \left( \frac{1}{1} \right) = 1 \cdot 1 = 1$$ Thus, $\lim_{x \to \infty} x \tan \left( \frac{1}{x} \right) = 1$.