Introduction to Limits in Calculus

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).

xf(x)
0.53
0.83.6
0.93.8
0.953.9
0.993.98
0.9993.998
0.99993.9998
0.999993.99998


We now consider x approaching 1 from the right (x > 1).

xf(x)
1.55
1.24.4
1.14.2
1.054.1
1.014.02
1.0014.002
1.00014.0002
1.000014.00002

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).

xg(x)
-0.50.9588
-0.20.993346
-0.10.998334
-0.010.999983
-0.0010.999999


xg(x)
0.50.9588
0.20.993346
0.10.998334
0.010.999983
0.0010.999999

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.

graph example 3

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.

graph example 4

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.

graph example 5

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.

graph example 6

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

graph example 7

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

graph example 8


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Updated: 2 April 2013

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