Introduction to Limits in Calculus

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_{xï؟½ 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_{xï؟½ 0} g(x) = 1. Note that g(0) is undefined.
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_{xï؟½ 1^-} f(x) = 2

As x approaches 1 from the right, y = f(x) approaches 4 and this can be written as

lim_{xï؟½ 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_{xï؟½ 1^-} f(x) = 2

As x approaches 1 from the right, y = f(x) approaches 4 and this can be written as

lim_{xï؟½ 1⁺} f(x) = 4

Note that the left hand limit and f(1) = 2 are equal.

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