Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?





a. limx→−1+ f(x) = 1

Verified step by step guidance

1. Examine the graph of the function y = f(x) as x approaches -1 from the right (x → -1+). This means you should look at the values of f(x) as x gets closer to -1 from values greater than -1.
2. Identify the y-value that the function approaches as x approaches -1 from the right. This is the value of the right-hand limit, lim_x→−1+ f(x).
3. Check if the y-value that the function approaches as x approaches -1 from the right is equal to 1.
4. If the y-value is equal to 1, then the statement lim_x→−1+ f(x) = 1 is true. Otherwise, it is false.
5. Conclude whether the statement is true or false based on the y-value identified in the previous steps.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity. For example, limx→−1+ f(x) refers to the limit of f(x) as x approaches -1 from the right side.

One-Sided Limits

One-sided limits are limits that consider the behavior of a function as the input approaches a specific value from one direction only. The notation limx→−1+ f(x) indicates the limit as x approaches -1 from the right (values greater than -1). This is crucial for analyzing functions that may behave differently from the left and right at a point.

Graphical Interpretation of Limits

Graphical interpretation of limits involves analyzing the graph of a function to determine the value that the function approaches as the input approaches a certain point. By observing the graph near x = -1, one can visually assess whether the function approaches a specific value, which aids in confirming or refuting statements about limits.