# To guess: The value of lim t → 0 e 5 t − 1 t .

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2.2, Problem 19E
To determine

## To guess: The value of limt→0e5t−1t.

Expert Solution

The limit of the function is guessed to 5.

### Explanation of Solution

Given:

The values of t are 0.5,0.1,0.01,0.001,0.0001,0.5,0.1,0.01,0.001 and 0.0001.

Calculation:

Since t approaches 0, the values 0.5,0.1,0.01,0.001, and0.0001 are to the left of 0 and the values 0.1,0.01,0.001 and 0.0001 are to the right of 3.

Evaluate the function (correct to 6 decimal places) when t=0.5,0.1,0.01,0.001, and0.0001 as shown in the below table.

 t e5t e5t−1 f(t)=e5t−1t −0.5 0.082084999 −0.91792 1.835830 −0.1 0.60653066 −0.39347 3.934693 −0.01 0.951229425 −0.04877 4.877057 −0.001 0.995012479 −0.00499 4.987520 −0.0001 0.999500125 −0.0005 4.998750

Here, the value of f(t) gets closer to the value 5 as t approaches 0 from the left side.

That is, limt0e5t1t=5.

Evaluate the function (correct to 6 decimal places) when t=0.5,0.1,0.01,0.001 and 0.0001 as shown in the below table.

 t e5t e5t−1 f(t)=e5t−1t 0.5 12.18249396 11.18249 22.364987 0.1 1.648721271 0.648721 6.487212 0.01 1.051271096 0.051271 5.127109 0.001 1.005012521 0.005013 5.012520 0.0001 1.000500125 0.0005 5.001250

Here, the value of f(t) gets closer to the value 5 as t approaches 0 from the right side.

That is, limt0+e5t1t=5.

Since the right hand limit and the left hand limits are the same, limt0e5t1t exists.

Therefore, the limit of the function is guessed to 5.

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