Question

In Exercises 67–70, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ f(c)=L, \text { then } \lim _{x \rightarrow c} f(x)=L $$

Short answer

The statement is false. A counterexample can be a function f(x) such that f(x) = 1 for x != 0 and f(0) = 2. It shows that f(c) can be different from \(\lim_{{x \to c}} f(x)\)

Step-by-step solution

Understanding the statement

The statement implies that the limit value of any function f(x) at a point x=c is the same as the function value at that point. The key to understanding this problem lies in grasping the concept of a limit. By definition, the limit of f(x) as x approaches c is L means that as the input values get arbitrarily close to the value c (but not equal to c), the output values get arbitrarily close to L.

Analyze the accuracy of the statement

The given statement says that if the function value at x=c is L (f(c)=L), then the limit as x approaches c is also L. This is an interesting assertion. However, it is important to note that the limit of a function as x approaches a particular value is all about the behavior of the function 'near' that value and isn't necessarily determined by the function's value 'at' that particular value. Thus, the statement can be false.

Giving an example

Consider a function f(x) which is defined as follows: f(x) = 1 for x != 0 and f(0) = 2. For x != 0, the function output gets arbitrarily close to 1 but at x=0, the function value is 2, not 1. Hence, this example shows that it can be the case that f(c)!=L while \(\lim_{{x \to c}} f(x)=L\).

Key concepts

Function Behavior

Understanding how a function behaves helps us predict its movements or tendencies. Function behavior focuses on what the function does for input values close to a specific point. For instance, if a function outputs values getting closer to a specific number as you input values closer to some particular x-value, this gives guidance on the function's tendency near that point. These tendencies—whether the function increases, decreases, jumps, or stays constant—are crucial for evaluating limits and understanding discontinuities.

• Behavior "near" a point: How the function acts as it approaches a certain x-value
• Tendencies tell us more about the function's nature than exact position

Point of Discontinuity

A point of discontinuity occurs when there's a break or gap in a function's graph at a specific x-value. It indicates a place where the function is not continuous. This can happen in several ways:

• A jump, where the function suddenly changes value
• A hole, where the function is undefined at that x-value even though the graph might continue on either side of it

Identifying these points helps us understand where a function cannot be described perfectly with a simple equation over an interval.

Limit Definition

The limit of a function is central to understanding how it behaves as the input approaches a certain value. The limit \(\lim_{{x \to c}} f(x)=L\) is defined as the value that f(x) gets closer to as x gets closer to c, but x is not exactly c. It is not about what happens at x=c and that’s a key point in solving such problems. In calculus, limits lay the foundation for defining derivatives and integrals, providing insights into the function's nature—especially where it might not be obvious or visible.

Limit Evaluation

Evaluating a limit means figuring out what value a function approaches as the input gets closer to a certain point. It requires understanding the function's behavior near the point, which may not coincide with the actual function value at the point itself. Methods to evaluate limits:

• Substitution: Directly substituting the x-value into the function if it’s continuous there
• Factoring: Simplifying the function by canceling out terms that cause undefined expressions like division by zero
• L'Hopital's Rule: Using derivatives when the limit results in indeterminate forms like 0/0

These methods help students confirm or disprove statements about a function's limits and train them to analyze better the function's tendencies even if it doesn’t show clear continuity.