WEBVTT

1
00:00:00.000 --> 00:00:05.700
Here is the problem: (x squared plus x minus 12)/(x squared minus 9).

2
00:00:06.151 --> 00:00:13.024
Find the limit of (x squared plus x minus 12) / (x squared minus 9) as x → 3.

3
00:00:13.824 --> 00:00:24.134
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
00:00:24.584 --> 00:00:32.003
Direct substitution. The expression is continuous at x equals 3, so the limit equals its value there.

5
00:00:32.803 --> 00:00:39.560
So the answer is Limit equals 7 over 6 approximately 1.166666667.

6
00:00:40.010 --> 00:00:54.906
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.

7
00:00:55.706 --> 00:01:25.137
Numerical values from the right agree: f(3.01) equals 1.1663894, f(3.001) equals 1.1666389, f(3.0001) equals 1.1666639, f(3.00001) equals 1.1666664, which checks out.

8
00:01:25.587 --> 00:01:54.264
Numerical values from the left agree: f(2.99) equals 1.1669449, f(2.999) equals 1.1666944, f(2.9999) equals 1.1666694, f(2.99999) equals 1.1666669, which checks out.