If the function f(x) = begin{cases} frac{cos | vedclass

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

Question

(B) For the function to be continuous at $x = 0$,we must have $\lim_{x 	o 0} f(x) = f(0) = -1$. Evaluating the limit: $\lim_{x 	o 0} \frac{\cos ax -

Vedclass · Accepted Answer

Arithmetic progression

Vedclass · Answer

(B) For the function to be continuous at $x = 0$,we must have $\lim_{x 	o 0} f(x) = f(0) = -1$. Evaluating the limit: $\lim_{x 	o 0} \frac{\cos ax - \cos bx}{\cos cx - \cos bx}$. Using the expansion $\cos 	heta \approx 1 - \frac{	heta^2}{2}$ for small $	heta$: $\lim_{x 	o 0} \frac{(1 - \frac{a^2x^2}{2}) - (1 - \frac{b^2x^2}{2})}{(1 - \frac{c^2x^2}{2}) - (1 - \frac{b^2x^2}{2})} = \lim_{x 	o 0} \frac{\frac{b^2x^2}{2} - \frac{a^2x^2}{2}}{\frac{b^2x^2}{2} - \frac{c^2x^2}{2}} = \frac{b^2 - a^2}{b^2 - c^2}$. Setting this equal to $-1$: $\frac{b^2 - a^2}{b^2 - c^2} = -1$. $b^2 - a^2 = -(b^2 - c^2) \implies b^2 - a^2 = c^2 - b^2$. $2b^2 = a^2 + c^2$. This condition implies that $a^2, b^2, c^2$ are in Arithmetic progression.

If the function $f(x) = \begin{cases} \frac{\cos ax - \cos bx}{\cos cx - \cos bx} & , x \neq 0 \\ -1 & , x = 0 \end{cases}$ is continuous at $x = 0$,then $a^2, b^2, c^2$ are in

Similar Questions

If $f(x) = \frac{x^2-10x+25}{x^2-7x+10}$ and $f$ is continuous at $x=5$,then $f(5)$ is equal to

Find all the points of discontinuity of the function $f$ defined by
$f(x) = \begin{cases} x + 2, & \text{if } x < 1 \\ 0, & \text{if } x = 1 \\ x - 2, & \text{if } x > 1 \end{cases}$

Let $f(x) = x^2 + x \sin x - \cos x$. Then

On the interval $I = [-2, 2]$,the function $f(x) = \begin{cases} (x + 1) e^{-\left[ \frac{1}{|x|} + \frac{1}{x} \right]} & x \neq 0 \\ 0 & x = 0 \end{cases}$ is given. Which one of the following does not hold good?

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module