Finding Limits Graphically


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





d. limx→1− f(x) = 2

Verified step by step guidance

1. To determine the limit \( \lim_{x \to 1^-} f(x) = 2 \), we need to analyze the behavior of the function \( f(x) \) as \( x \) approaches 1 from the left side (i.e., values of \( x \) that are slightly less than 1).
2. Examine the graph of the function \( y = f(x) \) and observe the values of \( f(x) \) as \( x \) gets closer to 1 from the left. Look for the y-values that the function approaches.
3. Identify the y-value that \( f(x) \) is approaching as \( x \) approaches 1 from the left. This is the left-hand limit, denoted as \( \lim_{x \to 1^-} f(x) \).
4. Check if the y-value that \( f(x) \) approaches from the left is equal to 2. If the graph shows that the function approaches a y-value of 2 as \( x \) approaches 1 from the left, then the statement is true.
5. If the y-value that \( f(x) \) approaches from the left is not equal to 2, then the statement is false. Conclude based on the observed behavior of the graph.

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 specific points, even if they are not defined at those points. For example, the limit of f(x) as x approaches 1 from the left (denoted as lim x→1− f(x)) examines the values f(x) takes as x gets closer to 1 from values less than 1.

One-Sided Limits

One-sided limits refer to the limits of a function as the input approaches a specific value from one side only. The left-hand limit (lim x→c− f(x)) considers values approaching c from the left, while the right-hand limit (lim x→c+ f(x)) considers values approaching c from the right. Understanding one-sided limits is crucial for analyzing functions that may have different behaviors on either side of 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 behavior of the graph near that point, one can visually assess whether the limit exists and what its value is. This method is particularly useful for identifying discontinuities or jumps in the function's values.