If f(x) = begin{cases} frac{a}{2}(x - |x|), & | vedclass

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

Question

(A) For $f(x)$ to be continuous at $x = 0$,we must have $\lim_{x 	o 0^{-}} f(x) = f(0) = \lim_{x 	o 0^{+}} f(x)$.First,consider the left-hand limit:

Vedclass · Accepted Answer

$a$ is any real value and $b$ is any real value

Vedclass · Answer

(A) For $f(x)$ to be continuous at $x = 0$,we must have $\lim_{x 	o 0^{-}} f(x) = f(0) = \lim_{x 	o 0^{+}} f(x)$.First,consider the left-hand limit: $\lim_{x 	o 0^{-}} f(x) = \lim_{x 	o 0^{-}} \frac{a}{2}(x - |x|)$.Since $x This holds true for any real value of $a$.Next,consider the right-hand limit: $\lim_{x 	o 0^{+}} f(x) = \lim_{x 	o 0^{+}} bx^2 \sin \left( \frac{1}{x} ight)$.We know that $-1 \leq \sin \left( \frac{1}{x} ight) \leq 1$.Multiplying by $bx^2$,we get $-|bx^2| \leq bx^2 \sin \left( \frac{1}{x} ight) \leq |bx^2|$.By the Squeeze Theorem,as $x 	o 0$,$bx^2 \sin \left( \frac{1}{x} ight) 	o 0$.This holds true for any real value of $b$.Thus,$f(x)$ is continuous at $x = 0$ for any real values of $a$ and $b$.

If $f(x) = \begin{cases} \frac{a}{2}(x - |x|), & \text{for } x < 0 \\ 0, & \text{for } x = 0 \\ bx^2 \sin \left( \frac{1}{x} \right), & \text{for } x > 0 \end{cases}$ is continuous at $x = 0$,then

Similar Questions

Examine the following function for continuity: $f(x) = \frac{x^{2} - 25}{x + 5}, x \neq -5$.

Let $f(x) = \begin{cases} (1 + |\sin x|)^{a/|\sin x|}, & -\pi/6 < x < 0 \\ b, & x = 0 \\ e^{\tan 2x/\tan 3x}, & 0 < x < \pi/6 \end{cases}$. If $f$ is continuous at $x = 0$,then the values of $a$ and $b$ are respectively:

If $f(x) = \begin{cases} x + \lambda, & x < 3 \\ 4, & x = 3 \\ 3x - 5, & x > 3 \end{cases}$ is continuous at $x = 3$,then $\lambda = $

If $f(x)$ is continuous on its domain $[-2,2]$,where $f(x) = \begin{cases} \frac{\sin ax}{x} + 3, & -2 \leq x < 0 \\ 2x + 7, & 0 \leq x \leq 1 \\ \sqrt{x^2+8} - b, & 1 < x \leq 2 \end{cases}$ then the value of $2a + 3b$ is

Consider the function $f(x) = \frac{x^3}{4} - \sin(\pi x) + 3$. Which of the following statements is true regarding the values attained by $f(x)$ in the interval $[-2, 2]$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module