If alpha = lim_{x rightarrow pi/4} frac{tan^{ | vedclass

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

Question

(D) First,calculate $\alpha = \lim_{x ightarrow \pi/4} \frac{	an^{3} x - 	an x}{\cos(x + \pi/4)}$. This is a $\frac{0}{0}$ form.Applying $L'H\hat{

Vedclass · Accepted Answer

$(1, 3)$

Vedclass · Answer

(D) First,calculate $\alpha = \lim_{x ightarrow \pi/4} \frac{	an^{3} x - 	an x}{\cos(x + \pi/4)}$. This is a $\frac{0}{0}$ form.Applying $L'H\hat{o}pital's$ rule:$\alpha = \lim_{x ightarrow \pi/4} \frac{3 	an^{2} x \sec^{2} x - \sec^{2} x}{-\sin(x + \pi/4)} = \frac{3(1)^{2}(2) - 2}{-\sin(\pi/2)} = \frac{6 - 2}{-1} = -4$.Next,calculate $\beta = \lim_{x ightarrow 0} (\cos x)^{\cot x}$. This is a $1^{\infty}$ form.$\beta = e^{\lim_{x ightarrow 0} \cot x (\cos x - 1)} = e^{\lim_{x ightarrow 0} \frac{\cos x - 1}{	an x}} = e^{\lim_{x ightarrow 0} \frac{-\sin x}{\sec^{2} x}} = e^{0} = 1$.Since $\alpha = -4$ and $\beta = 1$ are roots of $ax^{2} + bx - 4 = 0$,the sum of roots $\alpha + \beta = -b/a \Rightarrow -4 + 1 = -3 = -b/a \Rightarrow b = 3a$.The product of roots $\alpha \beta = -4/a \Rightarrow (-4)(1) = -4/a \Rightarrow a = 1$.Substituting $a = 1$ into $b = 3a$,we get $b = 3$.Thus,the ordered pair $(a, b)$ is $(1, 3)$.

If $\alpha = \lim_{x \rightarrow \pi/4} \frac{\tan^{3} x - \tan x}{\cos(x + \pi/4)}$ and $\beta = \lim_{x \rightarrow 0} (\cos x)^{\cot x}$ are the roots of the equation $ax^{2} + bx - 4 = 0$,then the ordered pair $(a, b)$ is:

Similar Questions

Evaluate the limit: $\lim _{x \rightarrow -9} \frac{(2.5)^{81-x^2}-(0.4)^{x+9}}{x+9}$

If $f(4) = g(4) = 2$,$f'(4) = 9$,and $g'(4) = 6$,then $\mathop {\text{Limit}}\limits_{x \to 4} \frac{\sqrt{f(x)} - \sqrt{g(x)}}{\sqrt{x} - 2}$ is equal to:

$\mathop {\lim }\limits_{x \to 0} \frac{{{e^{\tan x}} - {e^x}}}{{\tan x - x}} = $

$\mathop {\lim }\limits_{x \to \pi /4} \frac{{\sqrt 2 \cos x - 1}}{{\cot x - 1}} = $

The value of $\mathop {\lim }\limits_{x \to {0^ + }} {x^m}{(\log x)^n}$,where $m, n \in N$,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module