Lim x -1-28 x3-1-2 x-1

Lim_x →1/28x^3+1/2x+1 =8/2lim_x →1/2x^3 +(1/2)^3/x+1/2 =4 lim_x →1/2x^3 ( 1/2)^3/x ( 1/2) Applying formula lim_x → ax^n a^n/x a=na^n 1 Here, n=3, a= 1/2 =4 li ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x →1/28 x...

Question

${lim}_{x \to \frac{1}{2}} \frac{8 x^{3} + 1}{2 x + 1}$

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Solution

${lim}_{x \to \frac{1}{2}} \frac{8 x^{3} + 1}{2 x + 1}$

$= \frac{8}{2} {lim}_{x \to \frac{1}{2}} \frac{x^{3} + (\frac{1}{2})^{3}}{x + \frac{1}{2}}$

$= 4 {lim}_{x \to \frac{1}{2}} \frac{x^{3} - (- \frac{1}{2})^{3}}{x - (- \frac{1}{2})}$

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

Here, n=3, $a = \frac{- 1}{2}$

$= 4 {lim}_{x \to \frac{1}{2}} \frac{x^{3} - (- \frac{1}{2})^{3}}{(- \frac{1}{2})} = 4 \times 3 {(- \frac{1}{2})}^{3 - 1}$

$= 4 \times 3 \times \frac{1}{4} = 3$

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limx→128x3+12x+1

limx→128x3+12x+1 =82limx→12x3+(12)3x+12 =4limx→12x3−(−12)3x−(−12) Applying formula limx→axn−anx−a=nan−1 Here, n=3, a=−12 =4limx→12x3−(−12)3(−12)=4×3(−12)3−1 =4×3×14=3

${lim}_{x \to \frac{1}{2}} \frac{8 x^{3} + 1}{2 x + 1}$