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
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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
