Evaluate the limit-limx-0a-x2-a2x

Given:lim_x→0(a+x)^2 a^2x becomes 00 form Using Factorization method lim_x→0(a+x)^2 a^2x lim_x→0(a+x+a)(a+x a)x [∵ (a^2 b^2)=(a+b)(a b)] lim_x→0(2a+x)(x)x ...

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Standard XIII

Mathematics

Rationalization Method to Remove Indeterminate Form

Evaluate the ...

Question

Evaluate the limit: $lim x \to 0 \frac{{(a + x)}^{2} - a^{2}}{x}$

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Solution

Given: $lim x \to 0 \frac{{(a + x)}^{2} - a^{2}}{x}$ becomes $\frac{0}{0}$ form

Using Factorization method
$lim x \to 0 \frac{{(a + x)}^{2} - a^{2}}{x}$

$lim x \to 0 \frac{(a + x + a) (a + x - a)}{x}$

$[∵ (a^{2} - b^{2}) = (a + b) (a - b)]$

$lim x \to 0 \frac{(2 a + x) (x)}{x}$

$lim x \to 0 (2 a + x) = 2 a$

$T h e r e f o r e, \Rightarrow lim x \to 0 \frac{{(a + x)}^{2} - a^{2}}{x} = 2 a$

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Rationalization Method to Remove Indeterminate Form

Standard XIII Mathematics

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Evaluate the limit:limx→0(a+x)2−a2x

Given:limx→0(a+x)2−a2x becomes 00 form Using Factorization method limx→0(a+x)2−a2x limx→0(a+x+a)(a+x−a)x [∵(a2−b2)=(a+b)(a−b)] limx→0(2a+x)(x)x limx→0(2a+x)=2a Therefore,⇒limx→0(a+x)2−a2x=2a

Evaluate the limit: $lim x \to 0 \frac{{(a + x)}^{2} - a^{2}}{x}$