If lim _{x rightarrow 0}left(frac{cos 4 x+a c | vedclass

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

Question

(C) Let $f(x) = \cos 4x + a \cos 2x + b$. For the limit to be finite as $x \rightarrow 0$,the numerator must approach $0$ at a rate of at least $x^4$.

Vedclass · Accepted Answer

$-4, 3$

Vedclass · Answer

(C) Let $f(x) = \cos 4x + a \cos 2x + b$. For the limit to be finite as $x ightarrow 0$,the numerator must approach $0$ at a rate of at least $x^4$. Using the Taylor series expansion: $\cos 	heta = 1 - \frac{	heta^2}{2!} + \frac{	heta^4}{4!} - \dots$ $\cos 4x = 1 - \frac{(4x)^2}{2} + \frac{(4x)^4}{24} = 1 - 8x^2 + \frac{32}{3}x^4$ $a \cos 2x = a(1 - \frac{(2x)^2}{2} + \frac{(2x)^4}{24}) = a - 2ax^2 + \frac{2a}{3}x^4$ Substituting these into the numerator: $(1 + a + b) + (-8 - 2a)x^2 + (\frac{32}{3} + \frac{2a}{3})x^4$. For the limit to be finite,the coefficients of $x^0$ and $x^2$ must be zero. $1 + a + b = 0$ and $-8 - 2a = 0$. From $-8 - 2a = 0$,we get $a = -4$. Substituting $a = -4$ into $1 + a + b = 0$: $1 - 4 + b = 0 \implies b = 3$. Thus,the values are $a = -4, b = 3$.

If $\lim _{x \rightarrow 0}\left(\frac{\cos 4 x+a \cos 2 x+b}{x^4}\right)$ is finite,then the values of $a, b$ are respectively :

Similar Questions

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+a x^2+b x+c=0$,then $\alpha^{-1}+\beta^{-1}+\gamma^{-1}$ is equal to

Two concentric coils each of radius equal to $2\pi \, cm$ are placed at right angles to each other. $3\, A$ and $4\, A$ are the currents flowing in each coil respectively. The magnetic induction in $Wb/m^2$ at the centre of the coils will be $(\mu_0 = 4\pi \times 10^{-7} \, Wb/A \cdot m)$.

The mass of a planet is half that of the earth and the radius of the planet is one-fourth that of earth. If we plan to send an artificial satellite from the planet,the escape velocity will be,(escape velocity on earth $v_e = 11 \ km \ s^{-1}$) (in $km \ s^{-1}$)

$P, Q, R$ and $S$ are four points with the position vectors $3i - 4j + 5k, 2i - j + 4k, -4i + 5j + k$ and $-3i + 4j + 3k$, respectively. Then, the line $PQ$ meets the line $RS$ at the point

The hydride ion $H^{-}$ is a stronger base than the hydroxide ion $OH^{-}$. If sodium hydride $(NaH)$ is dissolved in water,which of the following reactions will occur?

Vedclass Test Series

Exam Paper Generator

Online Exam Module