Evaluate the limit-limx- 4x3-64x2-16

At x→ 4, x^3 64x^2 16 becomes 00 form Now, nbsp; lim_x→ 4x^3 64x^2 16 =lim_x→ 4x^3 4^3x^2 4^2 =lim_x→ 4 [ x^3 4^3x 4×x 4x^2 4^2 ] nbsp; nbsp; =lim_x ...

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Standard XII

Mathematics

Limit

Evaluate the ...

Question

Evaluate the limit:
$lim x \to 4 \frac{x^{3} - 64}{x^{2} - 16}$

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Solution

At $x \to 4$ ,

$\frac{x^{3} - 64}{x^{2} - 16}$ becomes $\frac{0}{0}$ form

Now,
$lim x \to 4 \frac{x^{3} - 64}{x^{2} - 16}$

$= lim x \to 4 \frac{x^{3} - 4^{3}}{x^{2} - 4^{2}}$

$= lim x \to 4 [\frac{x^{3} - 4^{3}}{x - 4} \times \frac{x - 4}{x^{2} - 4^{2}}]$

$= \frac{lim x \to 4 [\frac{x^{3} - 4^{3}}{x - 4}]}{lim x \to 4 [\frac{x^{2} - 4^{2}}{x - 4}]}$

$= \frac{3 (4)^{3 - 1}}{2 (4)^{2 - 1}} [∵ lim x \to a \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}]$

$= \frac{3 (4)^{2}}{2 (4)} = 6$

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Limit

Standard XII Mathematics

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Evaluate the limit: limx→4x3−64x2−16

At x→4, x3−64x2−16 becomes 00 form Now, limx→4x3−64x2−16 =limx→4x3−43x2−42 =limx→4[x3−43x−4×x−4x2−42] =limx→4[x3−43x−4]limx→4[x2−42x−4] =3(4)3−12(4)2−1[∵limx→axn−anx−a=nan−1] =3(4)22(4)=6

Evaluate the limit:
$lim x \to 4 \frac{x^{3} - 64}{x^{2} - 16}$