WEBVTT

1
00:00:00.000 --> 00:00:02.891
Here is the problem: (sin x)/x.

2
00:00:03.341 --> 00:00:07.834
Find the limit of (sin(x)) / x as x → 0.

3
00:00:08.634 --> 00:00:19.153
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
00:00:19.603 --> 00:00:20.485
Simplify.

5
00:00:20.935 --> 00:00:28.354
Direct substitution. The expression is continuous at x equals 0, so the limit equals its value there.

6
00:00:29.154 --> 00:00:31.441
So the answer is Limit equals 1.

7
00:00:31.891 --> 00:00:47.193
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
00:00:47.993 --> 00:01:14.406
Numerical values from the right agree: f(0.01) equals 0.99998333, f(0.001) equals 0.99999983, f(0.0001) equals 1, f(0.00001) equals 1, which checks out.

9
00:01:14.856 --> 00:01:42.116
Numerical values from the left agree: f(negative 0.01) equals 0.99998333, f(negative 0.001) equals 0.99999983, f(negative 0.0001) equals 1, f(negative 0.00001) equals 1, which checks out.