Limits
Limits of the Form 1^infinity
Premium Question
Grade 11
Question:
$\lim_{x \to 0} \left(\frac{1 + \tan x}{1 + \sin x}\right)^{\csc x}$
(1) $1$
(2) $e$
(3) $e^{1/2}$
(4) $e^{-1/2}$
Step-by-Step Solution
Key Concept: Take logarithm: $\csc x \cdot \ln \frac{1 + \tan x}{1 + \sin x} \approx \frac{\tan x - \sin x}{\sin x} \cdot 1 \text{ (near 1)} = \frac{\tan x - \sin x}{\sin x} = \frac{1}{\cos x} - 1 = \frac{1 - \cos x}{\cos x} \to 0$. So limit $= e^0 = 1$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)