Question

If $f(x)$ is continuous at point $x = 0$ where $f(x) = \begin{cases} \frac{3\sin x + 5\tan x}{\text{a}^x - 1} & , x<0 \\ \frac{2}{\log 2} & , x = 0 \\ \frac{8x + 2x\cos x}{\text{b}^x - 1} & , x>0 \end{cases}$ then the values of a and b, respectively, are ________

• A. 4, 5
• B. 16, 32
• C. 8, 10
• D. 16, 16

Hint:

Evaluate both one-sided limits at 0 and set each equal to the value of the function at 0.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Continuity at 0 needs left limit = right limit = \(f(0)=\frac{2}{\log2}\).

Step 2: Left limit
For negative x: \(\lim\frac{3\sin x+5\tan x}{a^x-1}=\frac{3+5}{\log a}=\frac{8}{\log a}\), using \(\frac{a^x-1}{x}\rightarrow\log a\).

Step 3: Find a
\(\frac{8}{\log a}=\frac{2}{\log2}\), so \(\log a=4\log2\) and \(a=16\).

Step 4: Right limit
For positive x: \(\lim\frac{8x+2x\cos x}{b^x-1}=\frac{8+2}{\log b}=\frac{10}{\log b}\).

Step 5: Find b
\(\frac{10}{\log b}=\frac{2}{\log2}\), so \(\log b=5\log2\) and \(b=32\).

Final Answer:
The values are \(a=16\) and \(b=32\). \[ \boxed{\text{(B) }16,\ 32} \]