Evaluating Limits from a Graph

In this discussion, we evaluate limits, including one-sided limits, from a graph.

Consider the graph of y = f(x).

For the examples in this discussion, we evaluate the following:

limx→0−f(x)

limx→0+f(x)

limx→0f(x)

limx→1−f(x)

limx→1+f(x)

limx→1f(x)

limx→2+f(x)

limx→2−f(x)

limx→2f(x)

limx→−1+f(x)

limx→−1−f(x)

limx→−1f(x)

Keep in mind that the x-values on the graph run from left to right and the y-values run from bottom to top.

Let’s Begin!

🔹Evaluate the Limit as x Approaches 0 From the Left

For the first question, we evaluate

limx→0−f(x)

Since x is approaching 0, I place one hand by x = 0 to indicate where to observe the graph horizontally.

Since x is approaching 0, in particularly, from the left, I place the other hand to the left of x = 0 and travel the graph, from left to right, until it is almost in vertical alignment with the first hand.

This is where I observe the y-value of the graph when the x-value approaches 0 from the left.

We see that the graph approaches 0 when x approaches 0 from the left.

Thus,

limx→0−f(x)=0

🔹Evaluate the Limit as x Approaches 0 From the Right

For the next question, we evaluate

limx→0+f(x)

Again, since x is approaching 0, I place one hand by x = 0 to indicate where to observe the graph horizontally.

Since x is approaching 0 from the right, I place the other hand to the right of x = 0 and travel the graph, from right to left, until it is almost in vertical alignment with the first hand.

This is where we observe the y-value of the graph when the x-value approaches 0 from the right.

We see that the graph approaches 0 when x approaches 0 from the right.

Thus,

limx→0+f(x)=0

🔹The Limit as x Approaches 0

Next, we evaluate

limx→0f(x)

Since the limit as x approaches 0 from the left is equal to the limit as x approaches 0 from the right, then the limit as x approaches 0 is equal to 0.

limx→0f(x)=0

Note that, although there is an open circle at x= 0, the limit still exists because a function doesn’t necessarily have to be defined at a point in order for the limit to exist at that point.

🔹The Limit as x Approaches 1 From the Left

For the next question, we evaluate

limx→1−f(x)

Since x is approaching 1, I place one hand by x = 1 to indicate where to observe the graph horizontally.

Since x is approaching 1 from the left, I place the other hand to the left of x = 1 and travel the graph, from left to right, until it is almost in vertical alignment with the first hand.

This is where we observe the y-value of the graph when x-value approaches 1 from the left.

We see that the graph approaches 1 when x approaches 1 from the left.

Thus,

limx→1−f(x)=1

🔹The Limit as x Approaches 1 From the Right

For the next question, we evaluate

limx→1+f(x)

Since x is approaching 1, I place one hand by x = 1 to indicate where to observe the graph horizontally.

Since x is approaching 1 from the right, I place the other hand to the right of x = 1 and travel the graph, from right to left, until it is almost in vertical alignment with the first hand.

This is where we observe the y-value of the graph when x-value approaches 1 from the right.

We see that the graph approaches 0 when x approaches 1 from the right.

Thus,

limx→1+f(x)=0

🔹The Limit as x Approaches 1

Next, we evaluate

limx→1f(x)

Since limit as x approaches 1 from the left does not equal the limit as x approaches 1 from the right, then the limit as x approaches 1 does not exist.

limx→1f(x)D.N.E.

🔹The Limit as x Approaches 2 From the Left.

For the next question, we evaluate

limx→2−f(x)

Since x is approaching 2, I place one hand by x = 2 to indicate where to observe the graph horizontally.

Since x is approaching 2 from the left, I place the other hand to the left of x = 2 and travel the graph, from left to right, until it is almost in vertical alignment with the first hand.

This is where we observe the y-value of the graph when x-value approaches 2 from the left.

We see that the graph approaches 0 when x approaches 2 from the left.

Thus,

limx→2−f(x)=0

🔹The Limit as x Approaches 2 From the Right

For the next question, we evaluate

limx→2+f(x)

Since 2 is a right endpoint, then the limit as x approaches 2 from the right does not exist because there are no values to the right of 2.

Thus,

limx→2+f(x)D.N.E

🔹The Limit as x Approaches 2

Since the limit as x approaches 2 from the left does not equal the limit as x approaches 2 from the right, then the limit as x approaches 2 does not exist.

limx→2f(x)D.N.E

🔹Evaluate the Limit as x Approaches -1 from the Left.

For the next question, we evaluate

limx→−1−f(x)

Since -1 is a left endpoint, then the limit as x approaches -1 from the left does not exist because there are no values to the left of -1.

Thus,

limx→−1−f(x)D.N.E.

🔹Evaluate the Limit as x Approaches-1 from the Right

Since x is approaching -1, I place one hand by x = -1 to indicate where to observe the graph horizontally.

Since x is approaching -1 from the right, I place the other hand to the right of x = -1 and travel the graph, from right to left, until it is almost in vertical alignment with the first hand.

This is where we observe the y-value of the graph when the x-value approaches -1 from the right.

We see that the graph approaches 1 when x approaches -1 from the right.

Thus,

limx→−1+f(x)=1

🔹Evaluate the Limit as x Approaches -1

Last, we evaluate

limx→−1f(x)

Since the limit as x approaches -1 from the left does not equal the limit as x approaches -1 from the right, then the limit as x approaches -1 does not exist.

limx→−1f(x)D.N.E

I hope my blog has been helpful.

There is more where that came from!

More on Limits

Finding Limits of Polynomial, Rational, and Radical Expressions

Proof of Lim x→0 sinx/x=1

Evaluating Limits of Trig Ratios

How do you find One Sided Limits Algebraically?

Evaluating Limits at Infinity

Understanding the Definition of the Derivative

Examples Using the Definition and Alternative Definition of the Derivative

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Angela Morgan