If lim x - ax9-a9-x-a-9- find all possible values of a-

Lim_x → ax^9 a^9/x a =9 ...(i) LHS =lim_x → ax^9 a^9/x a =9(a)^9 1 =9a^8 It is given that 9a^8 =9 [from (i)] ⇒ a^8 =9/9=1 ⇒ a^4 =1 a^2 =1 a = ± 1 ⇒ a =1 and a ...

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Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

If lim x → ax...

Question

If ${lim}_{x \to a} \frac{x^{9} - a^{9}}{x - a} = 9$ , find all possible values of a.

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Solution

${lim}_{x \to a} \frac{x^{9} - a^{9}}{x - a} = 9 . . . (i)$

$L H S = {lim}_{x \to a} \frac{x^{9} - a^{9}}{x - a}$

$= 9 (a)^{9 - 1}$

$= 9 a^{8}$

It is given that $9 a^{8} = 9$ [from (i)]

$\Rightarrow a^{8} = \frac{9}{9} = 1$

$\Rightarrow a^{4} = 1$

$a^{2} = 1$

$a = \pm 1$

$\Rightarrow a = 1$ and a =-1

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If limx→ax9−a9x−a=9, find all possible values of a.

limx→ax9−a9x−a=9...(i) LHS=limx→ax9−a9x−a =9(a)9−1 =9a8 It is given that 9a8=9 [from (i)] ⇒a8=99=1 ⇒a4=1 a2=1 a=±1 ⇒a=1 and a =-1

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