WEBVTT

1
00:00:00.000 --> 00:00:04.226
Here is the problem: (e to the power x minus 1)/x.

2
00:00:04.676 --> 00:00:10.295
Find the limit of (e to the power x minus 1) / x as x → 0.

3
00:00:11.095 --> 00:00:21.370
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
00:00:21.820 --> 00:00:22.691
Simplify.

5
00:00:23.141 --> 00:00:30.537
Direct substitution. The expression is continuous at x equals 0, so the limit equals its value there.

6
00:00:31.337 --> 00:00:33.636
So the answer is Limit equals 1.

7
00:00:34.086 --> 00:00:49.040
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
00:00:49.840 --> 00:01:18.575
Numerical values from the right agree: f(0.01) equals 1.0050167, f(0.001) equals 1.0005002, f(0.0001) equals 1.00005, f(0.00001) equals 1.000005, which checks out.

9
00:01:19.025 --> 00:01:49.861
Numerical values from the left agree: f(negative 0.01) equals 0.99501663, f(negative 0.001) equals 0.99950017, f(negative 0.0001) equals 0.99995, f(negative 0.00001) equals 0.999995, which checks out.