Question:
Evaluate:
$$
\lim_{x \to \infty} \left[ x - \log(\cosh x) \right]
$$
Evaluate:
$$
\lim_{x \to \infty} \left[ x - \log(\cosh x) \right]
$$
Show Hint
Use exponential approximation for large \( x \).
Updated On: Jun 4, 2025
- 2
- 0
- Not exist
- \(\log 2\)
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The Correct Option is D
Solution and Explanation
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: \[ x - \log(\cosh x) \approx x - (x - \log 2) = \log 2 \]
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