Evaluate the following limits-lim x - ax-25-2-a-25-2-x-a

Lim_x → a(x+2)^5/2 (a+2)^5/2/x a =lim_x → a(x+2)^5/2 (a+2)^5/2/(x+2) (a+2) Let y =x+2 and b =a+2 When x ⇒ a, Then x+2 → a+2 ⇒ y → b =lim_y → by^5/2 b^5/2/y b, = ...

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Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

Evaluate the ...

Question

Evaluate the following limits:

${lim}_{x \to a} \frac{(x + 2)^{\frac{5}{2}} - (a + 2)^{\frac{5}{2}}}{x - a}$

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Solution

${lim}_{x \to a} \frac{(x + 2)^{\frac{5}{2}} - (a + 2)^{\frac{5}{2}}}{x - a}$

$= {lim}_{x \to a} \frac{(x + 2)^{\frac{5}{2}} - (a + 2)^{\frac{5}{2}}}{(x + 2) - (a + 2)}$

Let y =x+2 and b =a+2

When $x \Rightarrow a,$

Then $x + 2 \to a + 2$

$\Rightarrow y \to b$

$= {lim}_{y \to b} \frac{y^{\frac{5}{2} - b^{\frac{5}{2}}}}{y - b},$

$= \frac{5}{2} b^{\frac{5}{2} - 1}$

[Using formula ${lim}_{x \to a} \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}]$

$= \frac{5}{2} (a + 2)^{\frac{5}{2} - 1}$

$= \frac{5}{2} (a + 2)^{\frac{3}{2}}$

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Evaluate the following limits: limx→a(x+2)52−(a+2)52x−a

limx→a(x+2)52−(a+2)52x−a =limx→a(x+2)52−(a+2)52(x+2)−(a+2) Let y =x+2 and b =a+2 When x⇒a, Then x+2→a+2 ⇒y→b =limy→by52−b52y−b, =52b52−1 [Using formula limx→axn−anx−a=nan−1] =52(a+2)52−1 =52(a+2)32

Evaluate the following limits:

${lim}_{x \to a} \frac{(x + 2)^{\frac{5}{2}} - (a + 2)^{\frac{5}{2}}}{x - a}$