Evaluate the limit by using a change of variable?

Question

12357 views around the world

You can reuse this answer
Creative Commons License

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#