Question

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

Answer 1

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`