Embibe Experts Solutions for Chapter: Limits, Exercise 1: Exercise 1

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 $f\left(x\right)=\frac{\sin \left(e^{x-2}-1\right)}{\ln \left(x-1\right)}$, then $\underset{x\to 2}{\lim }f\left(x\right)$ is equal to

If $\underset{x\to 0}{\lim }\left\{\frac{x\left(1+a\cos x\right)-b\sin x}{x^3}\right\}=1$, then

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

$\underset{x\to \frac{\pi }{2}}{\lim }\left(\frac{\sin x-{\left(\sin x\right)}^{\sin x}}{1-\sin x+\ln \left(\sin x\right)}\right)=$

What is $\underset{x\to \frac{\mathrm{\pi }}{6}}{\lim }\frac{2{\sin }^{2}x+\sin x-1}{2{\sin }^{2}x-3\sin x+1}$ equal to?