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One-sided Limits and Continuity
A limit written in the form of lim ( )
x a f x → is called a two-sided limit. This means that x is
approaching the number “a” from both sides (from the left and from the right). However, there may be times when you only want to find the limit from one side. To do this you would use one- sided limits.
One-sided limits are denoted by placing a positive (+) or negative (-) sign as an exponent on the value “a”. For example, if you wanted to find a one-sided limit from the left then the limit
would look like lim ( )
x a −^ f^ x →
. This limit would be read as “the limit of f(x) as x approaches a from
the left.”
A right-handed limit would look like lim ( )
x a +^ f^ x →
and would be read as “the limit of f(x) as
x approaches a from the right.”
Finding one-sided limits are important since they will be used in determining if the two- sided limit exists. For the two-sided limit to exist both one-sided limits must exist and be equal to the same value.
lim x → a f ( x ) exists if lim ( )
x a −^ f^ x →
= L and lim ( )
x a +^ f^ x^ M →
= and L = M.
The following three cases are situations where the limit of f as x approaches a may not exist.
- If f(x) approaches infinity (either positive or negative) as x approaches a from either side,
then the limit lim ( )
x a f x → does not exist.
lim ( ) x a − f^ x →
= ∞ lim ( ) x a
- If f(x) approaches positive infinity as x approaches a from one side and negative infinity
as x approaches a from the other side, then the limit lim ( )
x a f x → does not exist.
lim ( ) x a − f^ x →
= −∞ lim ( ) x a
- If f(x) approaches the number L from one side and the number M from the other side,
then the limit lim ( )
x a f x → does not exist.
lim ( ) x a −^ f^ x^ M →
= lim ( ) x a +^ f^ x^ L →
function is said to be continuous if there is no break (or gap) in the graph over an open ur r
A
interval. If you are able to sketch the graph of a function without having to stop and lift yo pencil from the graph then the function is continuous. However, if is not always convenient o possible to quickly sketch the graph of a function to determine if it is continuous at any given point. In order to determine if a function is continuous at a given point you would use the definition of continuity.
Example 1 (Continued):
Limit as x approaches –1 from the left is
1
lim ( ) 2 x − f^ x →−
Limit as x approaches –1 from the right is
1 lim ( ) 2 x
The left-handed limit does not approach the same number as the right- handed limit (-2 ≠ 2).
Therefore the function fails the second condition and is discontinuous at x = -1.
It is helpful to remember the characteristics of some of the more common graphs of basic functions. Keeping these characteristics in mind will help speed up the process of determining at what points (if any) a function is discontinuous.
Type of Function Sample Graph Continuity
Polynomial function
f ( x ) = a xn n^ + an − 1 x n −^1 + ...+ a x 1 + a 0 Continuous for all
values of x
Rational function
p x f x q x
Continuous for all values of x except for those that make the denominator zero
Square Root function
f ( x )= ax + b Continuous for all
values of x as long as the radicand is greater than or equal to zero.
Exponential function
f ( x ) = bx Continuous for all
values of x
Logarithmic function
f ( x ) = log bx Continuous for all
values of x greater than zero
Example 2: Find all values of x where the following function is discontinuous.
2 2
lim x 2
x x → x
Example 3 (Continued):
Solution:
The possible values of discontinuity for this piecewise function are at 1 and 2. So you would apply the tests of continuity for both of these values. First we will look at x = 1.
Test 1: f(1) is defined
When x is between 1 and 2 we will use the function f(x) = 2.
f(x) = 2 f(1) = 2
f(1) = 2 therefore the function is defined at x = 1 and passes the first test of continuity.
Test 2: ( )
1 lim x f x exists →
To determine if the limit exists we will compare the one-sided limits from the left and right to see if they approach the same value. Approaching 1 from the left we will use the function f(x) = 1 – x.
1 1
lim lim 1
1 1 0
x x −^ f^ x^ − x → →
Approaching 1 from the right we will use the function f(x) = 2.
1 1
lim lim 2
2
x x
Compare the one-sided limits
1 1 lim lim
x x −^ f^ x^ + f → → ≠ x
The one-sided limits do not approach the same value (0 ≠ 2) therefore the limit of the function as x approaches 1 does not exist. Since the limit does not exist we do not need to perform the third test and we can say the function is discontinuous at x = 1.
Example 3 (Continued):
Solution:
Now we will look at the second possible point of discontinuity, x = 2.
Test 1: f(2) is defined.
When x is between 1 and 2 we will use the function f(x) = 2.
f(x) = 2 f(2) = 2
f(2) = 2 therefore the function is defined at x = 2 and passes the first test of continuity.
Test 2: ( )
2 lim x f x exists →
First we will look at the limit approaching 2 from the left. For this limit will use the function f(x) = 2.
2 2
lim lim 2
2
x x − f^ x − → →
Now for the limit approaching 2 from the right we will use the function f(x) = 4 – x.
2 2
lim lim 4
4 2 2
x x +^ f^ x^ + x → →
Compare the one-sided limits
2 2
lim lim
x x −^ f^ x^ + f → →
= x
The one-sided limits approach the same value (2) therefore the limit of the function as x approaches 2 does exist and is equal to 2. So now we will look at the third and final test for continuity.
