LimitseasyFree

Limits — JEE Maths practice question

JEE Maths question with a full step-by-step solution.

Question
lim⁡x→∞cot⁡−1(x+1−x)sec⁡−1((2x+1x−1)x)\displaystyle\lim_{x\to\infty}\dfrac{\cot^{-1}\left(\sqrt{x+1}-\sqrt{x}\right)}{\sec^{-1}\left(\left(\dfrac{2x+1}{x-1}\right)^{x}\right)} is equal to
A11
B00
Cπ2\dfrac{\pi}{2}
DNon existent
Solution
Step 1: Compute x+1−x=1x+1+x→0\sqrt{x+1}-\sqrt{x}=\dfrac{1}{\sqrt{x+1}+\sqrt{x}}\to 0 as x→∞x\to\infty. So cot⁡−1(x+1−x)→cot⁡−1(0)=π2\cot^{-1}(\sqrt{x+1}-\sqrt{x})\to\cot^{-1}(0)=\dfrac{\pi}{2}. Step 2: Compute (2x+1x−1)x\left(\dfrac{2x+1}{x-1}\right)^{x}. As x→∞x\to\infty: 2x+1x−1→2\dfrac{2x+1}{x-1}\to 2, so (2x+1x−1)x→2∞=∞\left(\dfrac{2x+1}{x-1}\right)^{x}\to 2^{\infty}=\infty. So sec⁡−1(∞)=π2\sec^{-1}(\infty)=\dfrac{\pi}{2}. Step 3: Ratio: π/2π/2=1\dfrac{\pi/2}{\pi/2}=1. Correct answer: (1) 11
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Limits · easy
x x ( x+1 - x ) equal to
Limits · easy
x r=1 10 (x+r) 2010 (x 1006 +1)(2x 1004 +1) , is equal to
Limits · easy
x 0 ae x +b x+ce -x e 2x -2e x +1 =4 . Then
Limits · easy
The value of x 1 x 1/3 -x 1/4 x 3 -1 is

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath