How do you find #\lim _ { x \rightarrow 7} \frac { x ^ { 3} - 343} { x - 7}#?

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Answer 1 · 2018-01-27T03:36:59

#147#

Direct substitution does not work because it gives us 0/0, an indeterminate form.

We could try factorising this rational function so that we can possibly cancel out common terms.

The factorised numerator is: #(x-7)(x^2+7x+49)#
The denominator cannot be factorised further.

Therefore, we find that #lim_(xrarr7)(x^3-343)/(x-7)=lim_(xrarr7)((x-7)(x^2+7x+49))/(x-7)#.

We can cancel the #(x-7)#!

We get #lim_(xrarr7)x^2+7x+49#

If we substitute #x=7# into #x^2+7x+49#, we get:

#=7^2+7(7)+49#
#=49+49+49#
#=147#

This is the graph below.
graph{(x^3-343)/(x-7) [-24.6, 25.38, 129.5, 154.49]}