Embibe Experts Solutions for Chapter: Limits, Exercise 2: Level 2

If $\ell =\underset{\theta \to \frac{\pi }{4}}{\lim }\frac{\sqrt{2}-\cos \theta -\sin \theta }{{\left(4\theta -\pi \right)}^2}$, then find the value of $\frac{1}{\sqrt{2}\ell }$

Evaluate $\underset{x\to 0}{\mathrm{Lim}} \left[\ln \left(1+{\sin }^{2}x\right)·\cot \left({\ln }^2\left(1+x\right)\right)\right]$

If $\underset{x\to 0}{\lim }\left[\frac{x^a{\sin }^{b}x}{\sin \left(x^c\right)}\right]$, $a, b, c\in R-\left\{0\right\}$ exists and has non-zero value, then

If the value of $\underset{x\to 0}{\lim }{\left(2-\cos x\sqrt{\cos 2x}\right)}^{\left(\frac{x+2}{x^2}\right)}$ is equal to $e^a$, then $a$ is equal to_____.

The value of $\underset{x\to \infty }{\lim }\frac{{\left(2^{x^n}\right)}^{\frac{1}{e^x}}-{\left(3^{x^n}\right)}^{\frac{1}{e^x}}}{x^n}$(where $n\mathit{\in }N$) is

Let $f: [0,\infty )\to [0,\infty )$ be given by $f\left(x\right)=\sum _{i=1}^{10}\left(\frac{x^i}{i}\right)$. If $f^{\text{'}}\left(x\right)$ denotes the derivative of $f\left(x\right),$ then the value of $\underset{x\to 1}{\lim }\frac{f^{\text{'}}\left(x\right)-10}{x-1}$ is

Let $f\left(x\right)$ and $g\left(x\right)$ be differentiable functions on $\left(-\infty , \infty \right)$ and let $f^{\text{'}}\left(x\right)$ and $g^{\text{'}}\left(x\right)$ denote derivatives of $f\left(x\right)$ and $g\left(x\right)$, respectively. If $f\left(0\right)=\frac{1}{2}, g\left(0\right)=\frac{1}{3}, f^{\text{'}}\left(0\right)=1$ and $g^{\text{'}}\left(0\right)=2,$ then the value of $\underset{x\to 0}{\lim }\left[\frac{2f\left(2x^2+3x\right)-1}{3g\left(x\right)-1}\right]$ is

If for some real number $a,\underset{x\to 0}{\lim }\frac{\sin 2x+a\sin x}{x^3}$ exists, then the limit is equal to