WEBVTT

1
00:00:00.000 --> 00:00:03.669
Here is the problem: (1-cos(x))/x squared.

2
00:00:04.119 --> 00:00:09.193
Find the limit of (1 minus cos(x)) / x squared as x → 0.

3
00:00:09.993 --> 00:00:19.943
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
00:00:20.393 --> 00:00:30.691
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).

5
00:00:31.141 --> 00:00:32.012
Simplify.

6
00:00:32.462 --> 00:00:39.892
Direct substitution. The expression is continuous at x equals 0, so the limit equals its value there.

7
00:00:40.692 --> 00:00:45.290
So the answer is Limit equals 1 over 2 approximately 0.5.

8
00:00:45.740 --> 00:01:01.228
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.

9
00:01:02.028 --> 00:01:32.702
Numerical values from the right: f(0.01) equals 0.49999583, f(0.001) equals 0.49999996, f(0.0001) equals 0.5, f(0.00001) equals 0.50000004 (no clear trend at this precision).

10
00:01:33.152 --> 00:02:04.696
Numerical values from the left: f(negative 0.01) equals 0.49999583, f(negative 0.001) equals 0.49999996, f(negative 0.0001) equals 0.5, f(negative 0.00001) equals 0.50000004 (no clear trend at this precision).