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
- \(\log 2\)
Hide Solution
Verified By Collegedunia
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: \[ x - \log(\cosh x) \approx x - (x - \log 2) = \log 2 \]
Was this answer helpful?
0
0
Top Questions on Limits
- If \( f(3) = 18, f'(3) = 0 \) and \( f''(3) = 4 \), then the value of \[ \lim_{x \to 3} \ln \left( \frac{f(x+2)}{f(3)} \right)^{\frac{18}{(x-3)^3}} \] is equal to
- The limit of \( \lim_{x \to 0} \frac{\sin x}{x} \) is:
- Match List-I with List-II:
\[\begin{array}{|c|c|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline \text{(A)}\ \lim_{x\to 0}(1+2x)^{\frac{1}{x}} & \text{(I)}\ e^{6} \\ \hline \text{(B)}\ \lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x} & \text{(II)}\ e^{2} \\ \hline \text{(C)}\ \lim_{x\to 0}(1+5x)^{\frac{2}{x}} & \text{(III)}\ e \\ \hline \text{(D)}\ \lim_{x\to \infty}\left(1+\frac{3}{x}\right)^{2x} & \text{(IV)}\ e^{5} \\ \hline \end{array}\] Choose the correct answer from the options given below: - The value of $\displaystyle \lim_{x \to \infty}\left(1+\frac{2}{3x}\right)^{x}$ is:
- If $ \lim_{x \to 0} \frac{\cos(2x) + a \cos(4x) - b}{x^4} $ is finite, then $ (a + b) $ is equal to:
View More Questions
Questions Asked in AP EAPCET exam
- In a series LCR circuit, the voltages across the capacitor, resistor, and inductor are in the ratio 2:3:6. If the voltage of the source in the circuit is 240 V, then the voltage across the inductor is
- AP EAPCET - 2025
- Electromagnetic induction
- 0.25 moles of $ \text{CH}_2\text{FCOOH} $ was dissolved in $ 0.5 \, \text{kg} $ of water. The depression in freezing point of the resultant solution was observed as $ 1^\circ \text{C} $. What is the van't Hoff factor? ($ K_f = 1.86 \, \text{K kg mol}^{-1} $)
- AP EAPCET - 2025
- Colligative Properties
- At $T(K)$, the vapor pressure of water is $x$ kPa. What is the vapor pressure (in kPa) of 1 molal solution containing non-volatile solute?
- AP EAPCET - 2025
- Colligative Properties
- At 300 K, vapour pressure of pure liquid A is 70 mm Hg. It forms an ideal solution with liquid B. Mole fraction of B = 0.2 and total vapour pressure of solution = 84 mm Hg. What is vapour pressure (in mm) of pure B?
- AP EAPCET - 2025
- Colligative Properties
- A 1% (w/v) aqueous solution of a certain solute is isotonic with a 3% (w/v) solution of glucose (molar mass 180 g mol$^{-1}$). The molar mass of solute (in g mol$^{-1}$) is
- AP EAPCET - 2025
- Colligative Properties
View More Questions