Let f: R rightarrow R be defined as:f(x) = be | vedclass

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

Question

(C) For $f$ to be continuous at $x = 2$,we must have $\lim_{x \rightarrow 2^{-}} f(x) = \lim_{x \rightarrow 2^{+}} f(x) = f(2) = \mu$.First,calculate

Vedclass · Accepted Answer

$e(-e+1)$

Vedclass · Answer

(C) For $f$ to be continuous at $x = 2$,we must have $\lim_{x ightarrow 2^{-}} f(x) = \lim_{x ightarrow 2^{+}} f(x) = f(2) = \mu$.First,calculate the right-hand limit:$\lim_{x ightarrow 2^{+}} f(x) = \lim_{x ightarrow 2^{+}} e^{\frac{	an(x-2)}{x-[x]}}$. Since $x > 2$,$[x] = 2$,so $\lim_{x ightarrow 2^{+}} e^{\frac{	an(x-2)}{x-2}} = e^{1} = e$.Thus,$\mu = e$.Next,calculate the left-hand limit:$\lim_{x ightarrow 2^{-}} f(x) = \lim_{x ightarrow 2^{-}} \frac{\lambda|x^{2}-5x+6|}{\mu(5x-x^{2}-6)}$.Note that $x^{2}-5x+6 = (x-2)(x-3)$. For $x  0$. Thus,$|x^{2}-5x+6| = (x-2)(x-3)$.Also,$5x-x^{2}-6 = -(x^{2}-5x+6) = -(x-2)(x-3)$.So,$\lim_{x ightarrow 2^{-}} f(x) = \frac{\lambda(x-2)(x-3)}{\mu(-(x-2)(x-3))} = -\frac{\lambda}{\mu}$.Equating the limits: $-\frac{\lambda}{\mu} = e \Rightarrow \lambda = -\mu e = -e^{2}$.Therefore,$\lambda + \mu = -e^{2} + e = e(-e+1)$.

Similar Questions

Let $f(x) = \begin{cases} \frac{\tan^2 \{x\}}{x^2 - [x]^2} & \text{for } x > 0 \\ 1 & \text{for } x = 0 \\ \sqrt{\{x\} \cot \{x\}} & \text{for } x < 0 \end{cases}$ where $[x]$ is the greatest integer function and $\{x\}$ is the fractional part function of $x$,then:

The value that should be assigned to $f(0)$ so that the function $f(x)=(x+1)^{\cot x}$ is continuous at $x=0$, is

The values of $p$ and $q$ for which the function $f(x) = \begin{cases} \frac{\sin(p+1)x + \sin x}{x} & x < 0 \\ q & x = 0 \\ \frac{\sqrt{x+x^2} - \sqrt{x}}{x^{3/2}} & x > 0 \end{cases}$ is continuous for $\forall x \in R$ are

If $f(x) = \begin{cases} x \frac{e^{(1/x)} - e^{(-1/x)}}{e^{(1/x)} + e^{(-1/x)}}, & x \ne 0 \\ 0, & x = 0 \end{cases}$ then which of the following is true?

The number of discontinuities of the greatest integer function $f(x) = [x]$ for $x \in \left(-\frac{7}{2}, 100\right)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module