If f(x) = begin{cases} frac{sin((p+1)x) + sin | vedclass

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

Question

(D) For the function $f(x)$ to be continuous at $x = 0$,we must have $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^+} f(x) = f(0)$.First,calculate the Right

Vedclass · Accepted Answer

$(-3/2, 1/2)$

Vedclass · Answer

(D) For the function $f(x)$ to be continuous at $x = 0$,we must have $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^+} f(x) = f(0)$.First,calculate the Right Hand Limit $(RHL)$:$RHL = \lim_{x 	o 0^+} \frac{\sqrt{x+x^2} - \sqrt{x}}{x^{3/2}} = \lim_{x 	o 0^+} \frac{\sqrt{x}(\sqrt{1+x} - 1)}{x \cdot \sqrt{x}} = \lim_{x 	o 0^+} \frac{\sqrt{1+x} - 1}{x}$.Multiplying by the conjugate $\frac{\sqrt{1+x} + 1}{\sqrt{1+x} + 1}$,we get:$RHL = \lim_{x 	o 0^+} \frac{(1+x) - 1}{x(\sqrt{1+x} + 1)} = \lim_{x 	o 0^+} \frac{x}{x(\sqrt{1+x} + 1)} = \frac{1}{1+1} = \frac{1}{2}$.Next,calculate the Left Hand Limit $(LHL)$:$LHL = \lim_{x 	o 0^-} \frac{\sin((p+1)x) + \sin x}{x} = \lim_{x 	o 0^-} \left( \frac{\sin((p+1)x)}{x} + \frac{\sin x}{x} ight) = (p+1) + 1 = p+2$.Since $f(0) = q$,for continuity,we require $LHL = RHL = f(0)$:$p + 2 = 1/2 \implies p = 1/2 - 2 = -3/2$.$q = 1/2$.Thus,the ordered pair $(p, q) = (-3/2, 1/2)$.

If $f(x) = \begin{cases} \frac{\sin((p+1)x) + \sin x}{x} & , x < 0 \\ q & , x = 0 \\ \frac{\sqrt{x+x^2} - \sqrt{x}}{x^{3/2}} & , x > 0 \end{cases}$ is continuous at $x = 0$,then the ordered pair $(p, q)$ is equal to

Similar Questions

If $f(x) = \begin{cases} \frac{x^2 - 4x + 3}{x^2 - 1}, & x \ne 1 \\ 2, & x = 1 \end{cases}$,then:

If $f(x) = \lim_{n \rightarrow \infty} \left( \frac{\log(2+x) - x^{2n} \sin x}{1+x^{2n}} \right)$ for $0 \leq x \leq \frac{\pi}{2}$,then at $x=1$,$f(x)$ is

Find all points of discontinuity of $f,$ where $f$ is defined by $f(x) = \begin{cases} \frac{|x|}{x}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$

Given $f(x) = \begin{cases} cx + 1, & x \leq 3 \\ dx + 3, & x > 3 \end{cases}$. If $f$ is continuous at $x = 3$,then $d - c =$ . . . . . . .

Which one of the following functions is discontinuous at $x=1$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module