If f: R rightarrow R is defined by f(x) = beg | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) For a function $f(x)$ to be continuous at $x=0$, the condition $\lim_{x \rightarrow 0} f(x) = f(0)$ must hold. Given $f(x) = \frac{2 \sin x - \sin

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(D) For a function $f(x)$ to be continuous at $x=0$, the condition $\lim_{x ightarrow 0} f(x) = f(0)$ must hold. Given $f(x) = \frac{2 \sin x - \sin 2x}{2x \cos x}$ for $x 
eq 0$. We know that $\sin 2x = 2 \sin x \cos x$. Substituting this into the expression: $\lim_{x ightarrow 0} f(x) = \lim_{x ightarrow 0} \frac{2 \sin x - 2 \sin x \cos x}{2x \cos x}$ $= \lim_{x ightarrow 0} \frac{2 \sin x (1 - \cos x)}{2x \cos x}$ $= \lim_{x ightarrow 0} \left( \frac{\sin x}{x} ight) \cdot \left( \frac{1 - \cos x}{\cos x} ight)$ Since $\lim_{x ightarrow 0} \frac{\sin x}{x} = 1$ and $\lim_{x ightarrow 0} \frac{1 - \cos x}{\cos x} = \frac{1 - 1}{1} = 0$. Therefore, $\lim_{x ightarrow 0} f(x) = 1 \cdot 0 = 0$. Since $f(0) = a$, for continuity, we must have $a = 0$.

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{2 \sin x-\sin 2 x}{2 x \cos x}, & \text{if } x \neq 0 \\ a, & \text{if } x=0 \end{cases}$, then the value of $a$ so that $f$ is continuous at $x=0$ is

Similar Questions

Let $f:[-1,2] \rightarrow R$ be defined by $f(x)=[x^2-3]$ where $[\cdot]$ denotes the greatest integer function. The number of points of discontinuity for the function $f$ in the interval $(-1,2)$ is:

If the function $f(x)$,defined below,is continuous everywhere,then $k$ equals: $f(x)=\begin{cases} \frac{2^x-1}{\sqrt{1+x}-1}, & x \neq 0 \\ k, & x=0 \end{cases}$

If $f(x) = \begin{cases} e^{1/x}, & x \ne 0 \\ 0, & x = 0 \end{cases}$,then:

Given the function $f(x) = 2x \sqrt{x^3 - 1} + 5 \sqrt{x} \sqrt{1 - x^4} + 7x^2 \sqrt{x - 1} + 3x + 2$,then:

Let $f: R \rightarrow R$ be a continuous function satisfying $f(0)=1$ and $f(2x)-f(x)=x$ for all $x \in R$. If $\lim_{n \rightarrow \infty} \{f(x)-f(\frac{x}{2^n})\} = G(x)$,then $\sum_{r=1}^{10} G(r^2)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module