Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 1: Limits And Continuity – Section 1.1.4: One-Sided Limits

 

1.1.4
One-Sided Limits

 


Return To Contents
Go To Problems & Solutions

1. One-Sided Limits

 

and N, contradicting the uniqueness of limits; see Section 1.1.2 Theorem 2.1. However, although the limit of f at b
doesn't exist, its limits where only one side of b is taken into account exist. This situation has led to the concept of
one-sided limits.

 

Fig. 1.1


Go To Problems & Solutions     Return To Top Of Page

2. Left-Hand Limits

 

We say that a function f has the left-hand limit M as x approaches a, or that f approaches M as x approaches a from
the left, and we write:



if the domain of f contains points arbitrary close to but smaller than a and f(x) approaches arbitrarily close to L as x
approaches a from the left (x increases toward a).

In Fig. 1.1, f  has a left-hand limit of L at x = a and also a left-hand-limit of M at x = b, so that:



Example 2.1

 

 

Solution

EOS


As x approaches 2 from the left, it's clear that x3 approaches 8 and thus x3 – 1 approaches 7. The evaluation of left-hand
limits is the same as that of (two-sided) limits; see Section 1.1.2. Note that we say “x approaches a from the left ”, but we
don't say “f(x) approaches L from the left or below”; we don't care which side f(x) approaches L from.

 

Go To Problems & Solutions     Return To Top Of Page

3. Right-Hand Limits


We say that a function f has the right-hand limit L as x approaches a, or that f approaches L as x approaches a from
the right, and we write:



if the domain of f contains points arbitrary close to but greater than a and f(x) approaches arbitrarily close to L as x
approaches a from the right (x decreases toward a).

 

In Fig. 1.1, f  has a right-hand limit of L at x = a and also a right-hand-limit of N at x = b, so that:



Example 3.1

 

 

Solution

EOS


As x approaches 2 from the right, it's clear that x3 approaches 8 and thus x3 – 1 approaches 7. The evaluation of
right-hand limits is the same as that of (two-sided) limits; see Section 1.1.2. Note that we say “x approaches a from the
right ”, but we don't say “f(x) approaches L from the right or above”; we don't care which side f(x) approaches L from.

 

Go To Problems & Solutions     Return To Top Of Page

4. Evaluation

 

Right- and left-hand limits are referred to as one-sided limits. As seen in Examples 2.1 and 3.1, the evaluation of
one-sided limits is the same as that of (two-sided) limits; see Section 1.1.2.

 

Note that we say “x approaches a from the right ” or “x approaches a from the left ”, but we don't say “f(x) approaches L
from the right or above” or “f(x) approaches L from the left or below ”; we don't care which side f(x) approaches L from
or if it does so from both sides of L.

 

Go To Problems & Solutions     Return To Top Of Page

5. Limits And One-Sided Limits



Return To Top Of Page     Return To Top Of Page

6. Piecewise Functions

 

Example 6.1

 

Let:

 

 

Solution

1.

Fig. 6.1

 

Graph Of y = f(x).

 

EOS

 

Piecewise Functions

 

A piecewise function, also called a case-defined function, is a function whose graph consists of 2 or more pieces defined by different formulas, or by a single rule whose implementation changes with some subsets of the domain of the function, like the step function discussed in Problem & Solution 6. Direct substitution for limits, as discussed in Section 1.1.2 Part 4, isn't applicable to piecewise functions, as illustrated in Example 6.1 above.

 

A point of formula change is a point where the function changes formula. In this example, the points of formula change are x = –6, –2, and 3. To find the limit of a piecewise function at a point of formula change, we must consider both one-sided limits. This is because the formulas are different on each side. For a more subtle case of piecewise functions see Problem & Solution 6.

 

This example clearly demonstrates that the limit of a piecewise function at a point of formula change may or may not exist, and if it exists it may or may not be equal to the value of the function at that point if that value exists. So direct substitution for limits isn't applicable to piecewise functions at such points. To find limits at such points we must consider both one-sided limits, as previously stated.

 

Problems & Solutions


1.  Determine:


   


Solution



Return To Top Of Page



2.  Find:


   

 

    if it exists.

Solution

If x < 2, then x – 2 < 0, so |x – 2| = – (x – 2); thus:



Return To Top Of Page

 

3.  Find:


   

 

    if it exists.

Solution

From problem & solution 2 we get:

 


Return To Top Of Page

 

 

Solution

 


Return To Top Of Page

 

5.  Let:

 

    

 

Solution

 

a.

Graph Of y = f(x).

 

 

 

Return To Top Of Page

 

6.  Let [x] denote the greatest integer less than or equal to x.
     a.  Sketch a graph of [x]. Because of the look of its graph, this function is called a step function  and the notation [x] 
          can be read “step of x”, or, for short, “step x”.
    

 

Solution

 

a.

 

 


Return To Top Of Page     Return To Contents