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 lim_{x→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 lim_{x→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
lim_{x→1-} f(x) = 2
As x approaches 1 from the right, y = f(x) approaches 4 and this can be written as
lim_{x→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
lim_{x→1-} f(x) = 2
As x approaches 1 from the right, y = f(x) approaches 4 and this can be written as
lim_{x→1+} f(x) = 4
Note that the left hand limit and f(1) = 2 are equal.
**Example 5:**
This graph shows that
lim_{x→0-} f(x) = 1
and
lim_{x→0+} f(x) = 1
Note that the left and right hand limits are equal and we cvan write
lim_{x→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
lim_{x→-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
lim_{x→-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
lim_{x→ + ∞} 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
lim_{x→ - ∞ 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
lim_{x→ + ∞} f(x) = 2
If x is allowed to decrease without bound, f(x) approaches 2. This can be written
lim_{x→ - ∞} f(x) = 2
More on limits
Calculus Tutorials and Problems |