How do you find the limit lim_(x->9)(9-x)/(3-sqrt(x)) ?

1 Answer
Wataru
Sep 7, 2014

By cancelling common factors, we can find
lim_{x to 9}{9-x}/{3-sqrt{x}}=6.

Let us look at some details.
The first thing we should try when evaluating a limit is plug in the value. In this posted limit, we get 0/0 when we plug in x=9, which indicates that there should be a common factor (9-x) hidden in the expression. Since the factor (9-x) is already visible in the numerator, let us squeeze the factor out of the denominator.

By multiplying the numerator and the denominator by 3+sqrt{x} ,
lim_{x to 9}{9-x}/{3-sqrt{x}}cdot{3+sqrt{x}}/{3+sqrt{x}} =lim_{x to 9}{(9-x)(3+sqrt{x})}/{9-x}
by cancelling out (9-x),
=lim_{x to 9}(3+sqrt{x})=3+sqrt{9}=6

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