Question

Which of the following statements are correct? \[ \lim_{x \to 0} \frac{\sin x}{x} = 1,\quad \lim_{x \to 0} \frac{\tan x}{x} = 1 \]
• [A.] Limit exists and equals \(\frac{1}{\sqrt{2}}\)
• [B.] Limit does not exist
• [C.] Limit exists and equals \(\frac{1}{2\sqrt{2}}\)
• [D.] Limit exists and equals \(\frac{1}{2}\left(1 + \frac{1}{3}\right)\)
Choose the correct answer from the options given below:

• A. A, B, C only
• B. A, C only
• C. A, D only
• D. A, C, D only

Hint:

Always use standard limits and Taylor expansion for solving tricky trigonometric limits.

Answer 1

The Correct Option is D

Concept: Use standard trigonometric limits and series expansions.

Step 1: Use known limits. \[ \lim_{x \to 0} \frac{\sin x}{x} = 1,\quad \lim_{x \to 0} \frac{\tan x}{x} = 1 \]

Step 2: Expand using series. \[ \sin x = x - \frac{x^3}{6} + \cdots \] \[ \tan x = x + \frac{x^3}{3} + \cdots \]

Step 3: Substitute and simplify. After simplification, the expression reduces to a finite value. Thus limit exists.

Step 4: Evaluate given options.
• A: Correct
• B: Incorrect (limit exists)
• C: Correct
• D: Correct \[ \boxed{(4)} \]