Evaluate the limit by using a change of variable?

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Nelson

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Answer 1 · 2017-02-08T15:11:22

Let u = $(x+8)^(1/3)$

Then $u^3=x+8$ and $x = u^3-8$

As x approaches the value 0, u approaches the value 2. The given limit becomes

$lim_(x->0) ((x+8)^(1/3)-2)/x = lim_(u->2) (u-2)/(u^3-8)$

$(u-2)/((u-2)(u^2+2u+4))$

As $(u-2)$ cancels out and sub 2 in for u provides the final answer of $1/12$

Answer 2 · 2017-02-08T16:15:04

$1/12$

We observe that in the present form the limit becomes $0/0$ . Which is indeterminate.

Therefore, let us substitute
$(x+8)^(1/3)=u$
$=>x+8=u^3$
$=>x = u^3-8$

Also as $x->0$ , $u->2$

With this substitution the given question becomes

$lim_(u->2) (u-2)/(u^3-8)$

$=>lim_(u->2)(u-2)/((u-2)(u^2+2u+4))$
$=>lim_(u->2)1/((u^2+2u+4))$
$=>1/((2^2+2xx2+4))$
$=>1/12$