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Math 1314 Lesson 2 One-Sided Limits and Continuity
One-Sided Limits
Sometimes we are only interested in the behavior of a function when we look from one side and not from the other.
Example 1 : Consider the function x
x f ( x )=. Find lim ( ). 0 f x x →
Now suppose we are only interested in looking at the values of x that are bigger than 0. In this case, we are looking at a one-sided limit.
We write lim ( ) 0
f x x →+
. This is called a right-hand limit, because we are looking at values
on the right side of the target number.
In this case,
If we are interested in looking only at the values of x that are smaller than 0, then we would be finding the left-hand limit. The values of x that are smaller than 0 are to the left of 0 on the number line, hence the name. We write lim ( ) 0 f x x →−
In this case,
Our definition of a limit from the last lesson is consistent with this information. We say that f x L x a
→ lim ( ) , if and only if the function approaches the same value, L , from both the
left side and the right side of the target number. This idea is formalized in this theorem:
Theorem : Let f be a function that is defined for all values of x close to the target number a , except perhaps at a itself. Then f x L x a
→ lim ( ) if and only if lim f ( x ) lim f ( x ) L. x a x a
+ =^ −^ =
→ →
Example 2 : Consider this graph:
Find lim ( ),lim ( )andlim ( ) 0 0 0
f x f x f x x → −^ x →+ x →
, if it exists.
Continuity
We will be interested in finding where a function is continuous and where it is discontinuous. We’ll look at continuity over the entire domain of the function, over a given interval and at a specific point.
Continuity at a Point :
Here’s the general idea of continuity at a point: a function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. Stated a bit more formally,
A function f is said to be continuous at the point x = a if the following three conditions are met:
- f ( a ) is defined
- lim f ( x ) x → a exists
- lim f ( x ) x → a = f ( a )
You’ll need to check each of these three conditions to determine if a function is continuous at a specific point.
Example 5 : Determine if
x x
x x f x is continuous at x = 1.
If a function is not continuous at a point, then we say it is discontinuous at that point.
We find points of discontinuity by examining the function that we are given. A function can have a removable discontinuity, a jump discontinuity or an infinite discontinuity.
Example 6 : Find any points of discontinuity. State why the function is discontinuous at each point of discontinuity.
Continuity over an Interval
A function is continuous over the interval ( a, b ) if it is continuous at every point in the interval. We’ll state answers using interval notation.
Example 7 : Find the intervals on which f is continuous: 2
2
− −
x x
x x f x.
From this lesson you should be able to Say what we mean by a one-sided limits Find a one-sided limit from the graph of a function Find a one-sided limit from a piecewise-defined function Find a one-sided limit from a function Determine if a function is continuous at a point Find points of discontinuity over an interval or over the domain of the function given either a graph of the function or the function itself State intervals where a function is continuous given either a graph of the function or the function itself
