Question

Evaluate: $$ \lim_{x \to \infty} \left[ x - \log(\cosh x) \right] $$

• A. 2
• B. 0
• C. Not exist
• D. \(\log 2\)

Hint:

Use exponential approximation for large \( x \).

Answer 1

The Correct Option is D

Recall: \[ \cosh x = \frac{e^x + e^{-x}}{2} \] For large \( x \to \infty \): \[ \cosh x \approx \frac{e^x}{2} \] Then: \[ \log(\cosh x) \approx \log\left( \frac{e^x}{2} \right) = x - \log 2 \] So: \[ x - \log(\cosh x) \approx x - (x - \log 2) = \log 2 \]