Introductory Calculus: Limit of a Function and Continuity

The limit of a function from left or right or both:

If , we call the limit of this function as x approaches 1 from the left as

Similarly, we call the limit as x approaches 1 from the right equal to 2.

What is mysterious about this concept of limit?

The function obviously has a value of 2 when x = 1.

We say that

This is a simple thing to analyze because this function is continuous (the graph is “connected”) and we can clearly see that thefunction value at x = 1 is 2. See the graph below.

Suppose that

What is the limit as x approaches –1 from the left/right?

We will get two different values for this limit because there are two different expressions for .

If x approaches –1 from the left [for ], we use the formula

which gives us

If x approaches –1 from the right [for ], we use the formula

which gives us

In this case, we say that

Since these limits are different, we say that the ONE limit as x approaches –1 does not exist. In order for this limit to exist, both the left hand and right hand limits would have to be the same, and the graph would have to “connect” from the left and right sides.

These limits from the left and right have different values. Looking at a graph from a calculator screen, we can see that the left handgraph and the right hand graph do not meet in one point, but the limits from the left and right sides can be seen on the graph as the yvalues of this function for each piecewise-defined part of the graph. Notice the “open” circle and "closed" circle on the graph.

Notice that the value of the function at the point –1 is because only defines this function for the value x = –1. This is an important fact as we examine the continuity of a function. We will compare this value, if it exists, to the limit value.

Continuity of a function at a point x = a: Definition

A function is said to be continuous at a point where x = a if three conditions hold:

The limit of a function from left or right or both:xapproaches 1 from the left asSimilarly, we call the limit as

xapproaches 1 from the right equal to 2.The function obviously has a value of 2 when

x= 1.

We say thatThis is a simple thing to analyze because this function is continuous (the graph is “connected”) and we can clearly see that thefunction value at x = 1 is 2. See the graph below.

xapproaches –1 from the left/right?xapproaches –1from the left[for ], we use the formulaxapproaches –1from the right[for ], we use the formulaSince these limits are different, we say that the ONE limit asxapproaches –1 does not exist. In order for this limit to exist, both the left hand and right hand limits would have to be the same, and the graph would have to “connect” from the left and right sides.yvalues of this function for each piecewise-defined part of the graph. Notice the “open” circle and "closed" circle on the graph.functionat the point –1 is because only defines this function for the valuex= –1. This is an important fact as we examine the continuity of a function. We will compare this value, if it exists, to the limit value.Continuity of a function at a point x = a:Definition

we apply the definition of continuity at the point x = –1.x.

Find the following limits if they exist.||

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Graph and discuss continuity at x = 3.

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Graph and discuss continuity at x = 3.

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