Question

lim x—-a x^3-a^3/x^10-a^10

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:\displaystyle\lim_{x \to a}\rm \frac{ {x}^{3} - {a}^{3} }{ {x}^{10} - {a}^{10} }$

If we substitute directly x = a, we get

$\rm \:  =  \: \dfrac{ {a}^{3} - {a}^{3} }{ {a}^{10} - {a}^{10} }$

$\rm \:  =  \: \dfrac{0}{0}$

which is indeterminant form.

So,

$\rm :\longmapsto\:\displaystyle\lim_{x \to a}\rm \frac{ {x}^{3} - {a}^{3} }{ {x}^{10} - {a}^{10} }$

can be rewritten as

$\rm \:  =  \: \displaystyle\lim_{x \to a}\rm \dfrac{\dfrac{ {x}^{3} - {a}^{3} }{x - a} }{\dfrac{ {x}^{10} - {a}^{10} }{x - a} }$

$\rm \:  =  \: \dfrac{\displaystyle\lim_{x \to a}\rm \frac{ {x}^{3} - {a}^{3} }{x - a} }{\displaystyle\lim_{x \to a}\rm \frac{ {x}^{10} - {a}^{10} }{x - a} }$

We know

$\green{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{x \to a}\rm \frac{ {x}^{n} - {a}^{n} }{x - a} = {na}^{n - 1}}}}$

So, using this, we get

$\rm \:  =  \: \dfrac{ {3a}^{3 - 1} }{ {10a}^{10 - 1} }$

$\rm \:  =  \: \dfrac{ {3a}^{2} }{ {10a}^{9} }$

$\rm \:  =  \: \dfrac{3}{10} {a}^{2 - 9}$

$\rm \:  =  \: \dfrac{3}{10} {a}^{ - 7}$

$\rm \:  =  \: \dfrac{3}{10 {a}^{7} }$

Hence,

$\purple{\rm\implies \:\:\boxed{\tt{ \displaystyle\lim_{x \to a}\rm \frac{ {x}^{3} - {a}^{3} }{ {x}^{10} - {a}^{10} } = \frac{3}{10 {a}^{7} }}}}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

$\green{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{x \to 0}\rm \frac{sinx}{x} = 1}}}$

$\green{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{x \to 0}\rm \frac{tanx}{x} = 1}}}$

$\green{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{x \to 0}\rm \frac{log(1 + x)}{x} = 1}}}$

$\green{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{x \to 0}\rm \frac{ {e}^{x} - 1}{x} = 1}}}$

$\green{\rm :\longmapsto\:\boxed{\tt{ \displaystyle\lim_{x \to 0}\rm \frac{ {a}^{x} - 1}{x} = loga}}}$

Answer 2

$\color{red}{ \colorbox{blue}{\colorbox{orange} {hope \: it \: helps \: you}}}$