Evaluate lim x- a x-3 5-2 - a-3 5-2 x-a

lim_x→ a(x+3)^5/2 (a+3)^5/2x a multiply and divide by its conjugate =lim _x→ a((x+3)^5 (a+3)^5(x+3)^5/2+(a+3)^5/2x a) =lim _x→ a((x+3)^4+(x+3)^3(a ...

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Byju's Answer

Standard XII

Mathematics

Functions

Evaluate li...

Question

Evaluate $lim x \to a \frac{{(x + 3)}^{5 / 2} - {(a + 3)}^{5 / 2}}{x - a}$

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Solution

${lim}_{x \to a} \frac{(x + 3)^{\frac{5}{2}} - (a + 3)^{\frac{5}{2}}}{x - a}$

multiply and divide by its conjugate

$= {lim}_{x \to a} ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{\frac{{(x + 3)}^{5} - {(a + 3)}^{5}}{{(x + 3)}^{\frac{5}{2}} + {(a + 3)}^{\frac{5}{2}}}}{x - a} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$

$= {lim}_{x \to a} ⎛ ⎜ ⎝ \frac{{(x + 3)}^{4} + {(x + 3)}^{3} (a + 3) + {(x + 3)}^{2} {(a + 3)}^{2} + (x + 3) {(a + 3)}^{3} + {(a + 3)}^{4}}{{(x + 3)}^{\frac{5}{2}} + {(a + 3)}^{\frac{5}{2}}} ⎞ ⎟ ⎠$

$= \frac{{(a + 3)}^{4} + {(a + 3)}^{3} (a + 3) + {(a + 3)}^{2} {(a + 3)}^{2} + (a + 3) {(a + 3)}^{3} + {(a + 3)}^{4}}{{(a + 3)}^{\frac{5}{2}} + {(a + 3)}^{\frac{5}{2}}}$

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

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Evaluate limx→a(x+3)5/2−(a+3)5/2x−a

Evaluate $lim x \to a \frac{{(x + 3)}^{5 / 2} - {(a + 3)}^{5 / 2}}{x - a}$