MHT CET · Maths · Limits
\(\lim _{x \rightarrow 1} \frac{a b^x-a^x b}{x^2-1}=\)
- A \(\frac{-\mathrm{ab}}{2} \log \left(\frac{\mathrm{b}}{\mathrm{a}}\right)\)
- B \(\frac{\mathrm{ab}}{2} \log \left(\frac{\mathrm{b}}{\mathrm{a}}\right)\)
- C \(a b \log \left(\frac{b}{a}\right)\)
- D \(-a b \log \left(\frac{b}{a}\right)\)
Answer & Solution
Correct Answer
(B) \(\frac{\mathrm{ab}}{2} \log \left(\frac{\mathrm{b}}{\mathrm{a}}\right)\)
Step-by-step Solution
Detailed explanation
Let \(\lim _{x \rightarrow 1} \frac{a b^x-a^x b}{x^2-1}=L\)
\(
\begin{aligned}
& \Rightarrow L=\lim _{x \rightarrow 1} \frac{\left(a b^x \log b\right)-\left(a^x \log a \cdot b\right)}{2 x} \\
& =\frac{a b \log b-a b \log a}{2}=\frac{a b}{2} \log \left(\frac{b}{a}\right)
\end{aligned}
\)
... [L' Hospital rule]
\(
\begin{aligned}
& \Rightarrow L=\lim _{x \rightarrow 1} \frac{\left(a b^x \log b\right)-\left(a^x \log a \cdot b\right)}{2 x} \\
& =\frac{a b \log b-a b \log a}{2}=\frac{a b}{2} \log \left(\frac{b}{a}\right)
\end{aligned}
\)
... [L' Hospital rule]
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The line MN whose equation is \(x-y-2=0\) cuts the X-axis at M and co-ordinates of N are \((4,2)\). The line MN is rotated about M through \(45^{\circ}\) in anticlockwise direction. The equation of the line MN in the new position isMHT CET 2025 Medium
- The vectors \(\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{k}}+4 \hat{\mathrm{k}}\) are the sies of a triangle \(\mathrm{ABC}\). The length of the median through \(\mathrm{A}\) isMHT CET 2021 Easy
- Let \(P \equiv(-3,0), Q \equiv(0,0)\) and \(R \equiv(3,3 \sqrt{3})\) be three points. Then the equation of the bisector of the angle \(\mathrm{PQR}\) isMHT CET 2023 Medium
- The equation of the normal to the curve \(3 x^2-y^2=8\), which is parallel to the line \(x+3 y=10\), isMHT CET 2023 Easy
- If the lengths of the transverse axis and the latus rectum of a hyperbola are and respectively, then, the equation of the hyperbola is ________MHT CET 2019 Easy
- The eccentricity of the curve represented by \(x=3(\cos \mathrm{t}+\sin \mathrm{t})\), \(y=4(\cos t-\sin t)\) isMHT CET 2025 Medium
More PYQs from MHT CET
- Which of the following statements is NOT true?MHT CET 2024 Easy
- \(\int_0^1 \frac{8 \log (1+x)}{1+x^2} \mathrm{~d} x=\)MHT CET 2022 Easy
- If \(\sqrt{\log _3 x^{16}}+9 \log _{27} \sqrt[3]{\frac{3}{x}}=5\)
then \(x=\ldots\).MHT CET 2025 Medium - Which of the following reaction does NOT involve replacement of diazonium group?MHT CET 2020 Medium
- Match the structures/parts of human male reproductive system in Column I with their respective explanation in Column II and select the correct answer from given options.
Column I Column II i. Prepuce a. Fibromuscular band that retains testis in scrotum. ii. Penis b. Primary male reproductive gonad. iii. Testis c. Loose fold of skin covering glans penis. iv. Gubernaculum d. Male copulatory organ. MHT CET 2024 Hard - Organ of Corti is located on the ___________ .MHT CET 2017 Easy