mathop {lim }limits_{x to a} frac{{sqrt {3x - | vedclass

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

Question

(B) To evaluate the limit $\mathop {\lim }\limits_{x 	o a} \frac{{\sqrt {3x - a} - \sqrt {x + a} }}{{x - a}}$,we rationalize the numerator:$= \mathop

Vedclass · Accepted Answer

$1/\sqrt{2a}$

Vedclass · Answer

(B) To evaluate the limit $\mathop {\lim }\limits_{x 	o a} \frac{{\sqrt {3x - a} - \sqrt {x + a} }}{{x - a}}$,we rationalize the numerator:$= \mathop {\lim }\limits_{x 	o a} \frac{{\sqrt {3x - a} - \sqrt {x + a} }}{{x - a}} 	imes \frac{{\sqrt {3x - a} + \sqrt {x + a} }}{{\sqrt {3x - a} + \sqrt {x + a} }}$$= \mathop {\lim }\limits_{x 	o a} \frac{{(3x - a) - (x + a)}}{{(x - a)(\sqrt {3x - a} + \sqrt {x + a})}}$$= \mathop {\lim }\limits_{x 	o a} \frac{{2x - 2a}}{{(x - a)(\sqrt {3x - a} + \sqrt {x + a})}}$$= \mathop {\lim }\limits_{x 	o a} \frac{{2(x - a)}}{{(x - a)(\sqrt {3x - a} + \sqrt {x + a})}}$$= \frac{2}{{\sqrt {3a - a} + \sqrt {a + a}}} = \frac{2}{{\sqrt {2a} + \sqrt {2a}}} = \frac{2}{{2\sqrt {2a}}} = \frac{1}{{\sqrt {2a}}}$Alternatively,using $L'	ext{Hospital's rule}$:$= \mathop {\lim }\limits_{x 	o a} \frac{{\frac{d}{{dx}}(\sqrt {3x - a} - \sqrt {x + a})}}{{\frac{d}{{dx}}(x - a)}}$$= \mathop {\lim }\limits_{x 	o a} \frac{{\frac{3}{{2\sqrt {3x - a} }} - \frac{1}{{2\sqrt {x + a} }}}}{1}$$= \frac{3}{{2\sqrt {2a} }} - \frac{1}{{2\sqrt {2a} }} = \frac{2}{{2\sqrt {2a} }} = \frac{1}{{\sqrt {2a}}}$

$\mathop {\lim }\limits_{x \to a} \frac{{\sqrt {3x - a} - \sqrt {x + a} }}{{x - a}} = $

Similar Questions

$\mathop {\lim }\limits_{x \to 0} \frac{{{4^x} - {9^x}}}{{x({4^x} + {9^x})}} = $

$\lim _{x \rightarrow 0} \left( \frac{x \cdot 10^x - x}{1 - \cos x} \right)$ is equal to

If $f(x)=3 x^{15}-5 x^{10}+7 x^5+50 \cos (x-1)$,then $\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{h^3+3 h}=$

$\lim _{x \rightarrow 0} \frac{x^2 2^x-x^2 \sin x-x^2}{3^x+\cos x-3^x \cos x-1}=$

If $\lim _{x \rightarrow 0} \frac{ae^{x}-b \cos x + ce^{-x}}{x \sin x} = 2,$ then $a + b + c$ is equal to ...........

Vedclass Test Series

Exam Paper Generator

Online Exam Module