WEBVTT

1
00:00:00.000 --> 00:00:05.434
Here is the problem: (sqrt(x) minus 3)/(x minus 9).

2
00:00:05.884 --> 00:00:12.502
Find the limit of (sqrt(x) minus 3) / (x minus 9) as x → 9.

3
00:00:13.302 --> 00:00:24.006
Indeterminate form 0 over 0 — apply L prime Hôpital prime s rule. Differentiate the numerator and the denominator separately (this is not the quotient rule).

4
00:00:24.456 --> 00:00:25.327
Simplify.

5
00:00:25.777 --> 00:00:33.138
Direct substitution. The expression is continuous at x equals 9, so the limit equals its value there.

6
00:00:33.938 --> 00:00:40.974
So the answer is Limit equals 1 over 6 approximately 0.1666666667.

7
00:00:41.424 --> 00:00:56.784
A limit describes the value a function approaches, not necessarily the value it takes. When substitution gives an indeterminate form, the expression is rewritten (L prime Hôpital prime s rule, a conjugate, or logarithms) until substitution works.

8
00:00:57.584 --> 00:01:30.266
Numerical values from the right agree: f(9.01) equals 0.1666204, f(9.001) equals 0.16666204, f(9.0001) equals 0.1666662, f(9.00001) equals 0.16666662, which checks out.

9
00:01:30.716 --> 00:01:59.904
Numerical values from the left agree: f(8.99) equals 0.16671299, f(8.999) equals 0.1666713, f(8.9999) equals 0.16666713, f(8.99999) equals 0.16666671, which checks out.