Problem 103 - Calculus of a Single Variable

Question

True or False? In Exercises \(103-106,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\lim _{x \rightarrow c} f(x)=L\) and \(f(c)=L,\) then \(f\) is continuous at \(c\)

Step-by-Step Solution

Answer

The statement is True.

Step 1: Understanding the Definitions

The limit of a function at a particular point \(x = c\) is the value that the function approaches as \(x\) approaches \(c\). Continuity of function at a point implies that the limit of the function at that point is equal to the function of that point.

Step 2: Evaluating the Statement

Given that, \(\lim _{x \rightarrow c} f(x) = L\) and \(f(c) = L\). Hence, the condition of continuity of function at \(c\), \( \lim_{x \rightarrow c} f(x) = f(c)\), is fulfilled. Thus the function \(f\) is continuous at \(c\).

Key Concepts

Limits of a Function

The concept of limits is essential in calculus and describes the behavior of a function as it approaches a certain point. For instance, when we write \( \lim_{x \rightarrow c} f(x) = L \), we're saying that as \(x\) gets very close to \(c\), \(f(x)\) gets very close to some fixed value \(L\). This does not necessarily mean that \(f(x)\) equals \(L\) when \(x = c\), but rather that \(x\) can get as close to \(c\) as we want, and the function values will approach \(L\).

Understanding limits is crucial because it sets the foundation for determining the continuity of a function at a point, which has major implications in defining derivatives and integrating functions. For those new to calculus, it can initially be tricky to grasp, so relating to tangible examples like approaching a traffic light, which turns red as you get closer, might help.

Function Continuity at a Point

A function is said to be continuous at a point \(x = c\) if, and only if, three conditions are met: the function is defined at \(c\), the limit of the function as \(x\) approaches \(c\) exists, and the limit equals the function's value at \(c\). In mathematical terms, this is expressed as \( \lim_{x \rightarrow c} f(x) = f(c) \).

Why is continuity important? In practical terms, imagine drawing the graph of a function without lifting your pencil off the paper. If that's possible, the function is likely continuous. Discontinuities may occur because there is a jump, a hole, or an asymptote that prevents a smooth drawing. Understanding function continuity helps students in graphing functions, predicting function behavior, and it is essential in advanced applications like integrals and series.

True or False Exercises in Calculus

True or false exercises serve as a great educational tool to test understanding of calculus concepts. They force students to not only recognize if a statement is correct but also understand why. When faced with a statement such as 'If \(\lim _{x \rightarrow c} f(x)=L\) and \(f(c)=L\), then \(f\) is continuous at \(c\)', it's crucial to evaluate this against the definition of continuity. In this case, the statement is indeed True. If a student finds it false, it's an opportunity to revisit the definitions of limits and function continuity. Additionally, if a statement is false, explaining why or providing a counterexample solidifies the learning process and enhances critical thinking skills, which are vital in the field of calculus.