How To Evaluate Limits From a Graph

Summary

This video tutorial explains how to evaluate limits from a graph, covering one-sided and two-sided limits, function values, and continuity. It demonstrates identifying limits as x approaches a point from the left and right, including cases where limits do not exist due to mismatched values or vertical asymptotes. The video also explores limits at infinity using horizontal asymptotes and clarifies the distinction between the limit of a function and the function's actual value at a point. Through multiple examples with open and closed circles, vertical asymptotes, and horizontal asymptotes, viewers learn a step-by-step approach to interpreting graphs and determining limits accurately.

Takeaways

• 😀 A one-sided limit is the value the function approaches as x approaches a specific value from either the left or the right side.
• 😀 When the left-side and right-side limits are different, the overall limit does not exist.
• 😀 The value of f(x) at a specific x value is determined by looking for the closed circle at that point on the graph.
• 😀 If the limit as x approaches a value from both sides is the same, the limit exists and is equal to that value.
• 😀 When there is no closed circle at x = a, the function is not defined at that point, meaning f(a) does not exist.
• 😀 Asymptotes (vertical or horizontal) can influence how limits behave as x approaches infinity or negative infinity.
• 😀 A function may approach positive or negative infinity as x approaches a certain value, indicating unbounded growth or decay.
• 😀 Infinity is not a specific number, but a concept describing the behavior of a function as it grows without bound.
• 😀 Horizontal asymptotes indicate the value that a function approaches as x goes to positive or negative infinity.
• 😀 Vertical asymptotes represent values where the function grows infinitely in either direction as x approaches that point.

Q & A

What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit refers to the behavior of a function as it approaches a certain point from either the left or the right side, while a two-sided limit considers the behavior from both sides simultaneously. A two-sided limit exists only when both one-sided limits are equal.

How do you determine the value of a limit from a graph?

To find the value of a limit from a graph, examine the y-values the function approaches as x approaches the specified point from both the left and the right sides. If both sides match, the limit exists. If they don't, the limit does not exist.

What happens when the left-hand and right-hand limits are not equal?

When the left-hand limit and right-hand limit are not equal, the two-sided limit does not exist. This means the function does not approach a specific value as x approaches the given point from both sides.

How do you find the value of f(x) at a specific point from a graph?

To find the value of f(x) at a specific point, locate the closed circle at the given x-value on the graph. The y-coordinate of the closed circle represents the value of f(x) at that point. If there's no closed circle, f(x) is not defined at that point.

What does an open circle on a graph indicate in terms of limits?

An open circle on a graph indicates that the function is not defined at that specific point. However, the limit as x approaches that point from either side may still exist, depending on the behavior of the function near the point.

How do you evaluate limits involving infinity?

To evaluate limits involving infinity, look at the behavior of the function as x approaches very large (positive or negative) values. If the function approaches a horizontal asymptote, the limit will be equal to the y-value of that asymptote.

What does it mean when a function has a vertical asymptote at x = 3?

A vertical asymptote at x = 3 means that as x approaches 3 from either side, the function's values become arbitrarily large (positive or negative). The limit does not exist in this case, since the function diverges to infinity.

What is the significance of a closed circle in a graph when finding a limit?

A closed circle in a graph indicates that the function is defined at that point. When determining the value of f(x) at that point, you refer to the y-value of the closed circle. It also affects the evaluation of limits at that point if the function is continuous.

How can you identify the limit at infinity on a graph with a horizontal asymptote?

To identify the limit at infinity, observe the behavior of the graph as x approaches very large or very small values. The function will approach a constant y-value if there is a horizontal asymptote, and that y-value will be the limit as x approaches positive or negative infinity.

What does it mean when the limit at a point is said to 'not exist'?

The limit at a point is said to 'not exist' when the function's behavior as x approaches the point from the left and right sides is not the same, or if the function diverges to infinity or does not approach any specific value. This indicates that the function does not have a well-defined limit at that point.