We intend to give a numerical and graphical approaches to the concept of limits using examples.
Numerical Approach to Limits
Example 1: Let f(x) = 2 x + 2 and compute f(x) as x takes values closer to 1. We first consider values of x approaching 1 from the left (x < 1).
| x | f(x) |
| 0.5 | 3 |
| 0.8 | 3.6 |
| 0.9 | 3.8 |
| 0.95 | 3.9 |
| 0.99 | 3.98 |
| 0.999 | 3.998 |
| 0.9999 | 3.9998 |
| 0.99999 | 3.99998 |
We now consider x approaching 1 from the right (x > 1).
| x | f(x) |
| 1.5 | 5 |
| 1.2 | 4.4 |
| 1.1 | 4.2 |
| 1.05 | 4.1 |
| 1.01 | 4.02 |
| 1.001 | 4.002 |
| 1.0001 | 4.0002 |
| 1.00001 | 4.00002 |
In both cases as x approaches 1, f(x) approaches 4. Intuitively, we say that limx� 1
f(x) = 4.
NOTE: We are talking about the values that f(x) takes when x gets closer to 1 and not f(1). In fact we may talk about the limit of f(x) as x approaches a even when f(a) is undefined.
Example 2: Let g(x) = sin x / x and compute g(x) as x takes values closer to 0. We consider values of x approaching 0 from the left (x < 0) and values of x approaching 0 from the right (x > 0).
| x | g(x) |
| -0.5 | 0.9588 |
| -0.2 | 0.993346 |
| -0.1 | 0.998334 |
| -0.01 | 0.999983 |
| -0.001 | 0.999999 |
| x | g(x) |
| 0.5 | 0.9588 |
| 0.2 | 0.993346 |
| 0.1 | 0.998334 |
| 0.01 | 0.999983 |
| 0.001 | 0.999999 |
Here we say that limx� 0
g(x) = 1. Note that g(0) is undefined.
Graphical Approach to Limits
Example 3:
The graph below shows that as x approaches 1 from the left, y = f(x) approaches 2 and this can be written as
limx� 1- f(x) = 2
As x approaches 1 from the right, y = f(x) approaches 4 and this can be written as
limx� 1+ f(x) = 4
Note that the left and right hand limits and f(1) = 3 are all different.
Example 4:
This graph shows that
limx� 1- f(x) = 2
As x approaches 1 from the right, y = f(x) approaches 4 and this can be written as
limx� 1+ f(x) = 4
Note that the left hand limit and f(1) = 2 are equal.
Example 5:
This graph shows that
limx� 0- f(x) = 1
and
limx� 0+ f(x) = 1
Note that the left and right hand limits are equal and we cvan write
limx� 0 f(x) = 1
In this example, the limit when x approaches 0 is equal to f(0) = 1.
Example 6:
This graph shows that as x approaches - 2 from the left, f(x) gets smaller and smaller without bound and there is no limit. We write
limx� -2- f(x) = - �
As x approaches - 2 from the right, f(x) gets larger and larger without bound and there is no limit. We write
limx� -2+ f(x) = + �
Note that - � and � are symbols and not numbers. These are symbols used to indicate that the limit does not exist.
Example 7:
The graph below shows a periodic function whose range is given by the interval [-1 , 1]. If x is allowed to increase without bound, f(x) take values within [-1 , 1] and has no limit. This can be written
limx� + � f(x) = does not exist
If x is allowed to decrease without bound, f(x) take values within [-1 , 1] and has no limit again. This can be written
limx� - � f(x) = does not exist
Example 8:
If x is allowed to increase without bound, f(x) in the graph below approaches 2. This can be written
limx� + � f(x) = 2
If x is allowed to decrease without bound, f(x) approaches 2. This can be written
limx� - � f(x) = 2
More on limits
Calculus Tutorials and Problems
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