Let [t] represent the greatest integer not ex | vedclass

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

Question

(C) Given $f(x)$ is continuous at $x=2$,we have $\lim_{x 	o 2^-} f(x) = f(2)$.$\lim_{x 	o 2^-} (ae^x + [x-2]) = [e^{-2}] - C$$ae^2 + [0^-] = 0 - (1

Vedclass · Accepted Answer

$x=0$ only

Vedclass · Answer

(C) Given $f(x)$ is continuous at $x=2$,we have $\lim_{x 	o 2^-} f(x) = f(2)$.$\lim_{x 	o 2^-} (ae^x + [x-2]) = [e^{-2}] - C$$ae^2 + [0^-] = 0 - (1 - 2e^2)$$ae^2 - 1 = -1 + 2e^2 \Rightarrow a = 2$.Now,check continuity at $x=0$:$LHL = \lim_{x 	o 0^-} [e^x] = [e^0] = [1] = 1$.$RHL = \lim_{x 	o 0^+} (2e^x + [x-2]) = 2(1) + [-2] = 2 - 2 = 0$.Since $LHL 
eq RHL$,$f(x)$ is discontinuous at $x=0$.Check continuity at $x=1$:$f(1) = 2e^1 + [1-2] = 2e - 1$.$\lim_{x 	o 1^-} f(x) = 2e - 1$.$\lim_{x 	o 1^+} f(x) = 2e + [1-2] = 2e - 1$.Since $LHL = RHL = f(1)$,$f(x)$ is continuous at $x=1$.Thus,$f(x)$ is discontinuous only at $x=0$.

Let $[t]$ represent the greatest integer not exceeding $t$ and $C=1-2e^2$. If the function $f(x)=\begin{cases} [e^x], & x < 0 \\ ae^x+[x-2], & 0 \leq x < 2 \\ [e^{-x}]-C, & x \geq 2 \end{cases}$ is continuous at $x=2$,then $f(x)$ is discontinuous at

Similar Questions

Discuss the continuity of the function $f$,where $f$ is defined by $f(x) = \begin{cases} -2, & \text{if } x \le -1 \\ 2x, & \text{if } -1 < x \le 1 \\ 2, & \text{if } x > 1 \end{cases}$. Is it continuous at $x=3$?

If the function $f(x) = \begin{cases} -2 \sin x, & x \leq \frac{-\pi}{2} \\ A \sin x+B, & \frac{-\pi}{2} < x < \frac{\pi}{2} \\ \cos x, & x \geq \frac{\pi}{2} \end{cases}$ is continuous everywhere,then the values of $A$ and $B$ are respectively

Let $f, g: R \to R$ be two functions defined by $f(x) = \begin{cases} x \sin \left( \frac{1}{x} \right), & x \ne 0 \\ 0, & x = 0 \end{cases}$ and $g(x) = x f(x)$.
Statement $I$: $f$ is a continuous function at $x = 0$.
Statement $II$: $g$ is a differentiable function at $x = 0$.

If $\lim _{x \rightarrow a^{+}} f(x)=p, \lim _{x \rightarrow a^{-}} f(x)=m$ and $f(a)=k$,then which one of the following is true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module