Finding Limits Graphically


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





c. limx→0− f(x) = 0

Verified step by step guidance

1. To determine the limit \( \lim_{x \to 0^-} f(x) \), we need to analyze the behavior of the function \( f(x) \) as \( x \) approaches 0 from the left side (negative side).
2. Examine the graph of the function \( y = f(x) \) and observe the values of \( f(x) \) as \( x \) gets closer to 0 from the left.
3. Identify the y-value that \( f(x) \) approaches as \( x \) approaches 0 from the left. This is the value of the left-hand limit.
4. If the y-value that \( f(x) \) approaches is 0, then the statement \( \lim_{x \to 0^-} f(x) = 0 \) is true. Otherwise, it is false.
5. Conclude whether the statement is true or false based on the observed behavior of the function on 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 0 from the left (denoted as lim x→0− f(x)) examines the values f(x) takes as x gets closer to 0 from negative values.

One-Sided Limits

One-sided limits refer to the limits of a function as the input approaches a specific value from one side only, either the left or the right. The notation lim x→a− f(x) indicates the limit as x approaches 'a' from the left, while lim x→a+ f(x) indicates the limit 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 specific point. By observing the behavior of the graph near that point, one can infer the limit. This visual approach is particularly useful for identifying discontinuities or jumps in the function, which can affect the existence of limits.