Question

find the limit of (x^m-a^m) /(x^n-a^n) limit of x is a

Answer 1

hope this hlppp............

Answer 2

$\bold{ANSWER}$

$\frac{m}{n}\times\:a^{m-n}$

$\mathbb{EXPLANATION}$

$\rm{put\:x=0\:in\:given\:limit\:we\:get\:\frac{0}{0}\:form}$

$\rm{Now\:inorder\:to\:get\:rid\:from\:this\:form\:we\:use\:LHOSPITALS\:RULE}$

$\underline{L'HOSPITALS\:RULE}$

$\rm{it\:states\:that\:whenever\:we\:have\:\frac{0}{0}\:form\:We\: Differentiate\:Numenator}$

$\rm{And\:Denominator\:separately\:till\:we\:come\:out\:from\frac{0}{0}}$

$\displaystyle\lim_{x\to\:a}\frac{x^m-a^n}{x^n-a^n}$

$\underline{Differentiate\:Numerator\:And\:Denominator\:Separately}$

$\displaystyle\lim_{x\to\:a}\frac{mx^{m-1}-0}{nx^{n-1}-0}$

$\displaystyle\lim_{x\to\:a}\frac{mx^{m-1}}{nx^{n-1}}$

$\displaystyle\lim_{x\to\:a}\:\frac{m}{n}\times\:x^{m-n}$

$\frac{m}{n}\times\:a^{m-n}$

$\therefore$

$\displaystyle\lim_{x\to\:a}\frac{x^m-a^m}{x^n-a^n}$=$\frac{m}{n}\times\:a^{m-n}$