Question

True or False? In Exercises \(115-120\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=g(x)\) for all real numbers other than \(x=0,\) and \(\lim _{x \rightarrow 0} f(x)=L,\) then \(\lim _{x \rightarrow 0} g(x)=L\)

Short answer

The statement is false.

Step-by-step solution

Understand the statement

The given statement says that if we have two functions \(f(x)\) and \(g(x)\) that are equal for all real numbers except at \(x = 0\). It's also given that if \(\lim _{x \rightarrow 0} f(x) = L\), then \(\lim _{x \rightarrow 0} g(x) = L\). We have to check the validity of this statement.

Counter example

Let's consider two functions \(f(x) = x\) and \(g(x) = x\) when \(x ≠ 0\) and \(g(0) = 1\). Here \(f(x) = g(x)\) for all real numbers other than \(x = 0\). But, while \(\lim _{x \rightarrow 0} f(x) = 0\), \(\lim _{x \rightarrow 0} g(x) = 1\). Hence, the given statement is false.

Key concepts

Function Equality

In mathematical terms, function equality means that two functions, say \( f(x) \) and \( g(x) \), have identical values for every input within their domains. They must produce the same outputs for all elements where both are defined.
Here's what this might look like in practice:

• If \( f(x) = 2x \) and \( g(x) = 2x \), then clearly \( f(x) = g(x) \) for every \( x \).
• If two functions differ at even one point, they are not considered equal over that range.

Consider a scenario where two functions are supposed to be equal except at some isolated points, like in the exercise at \( x = 0 \). Such exceptions can lead to different outcomes when analyzing their limits, illustrating how function equality is crucial in the broader context of calculus.

Limit at a Point

In calculus, a limit at a point describes the value that a function approaches as its input gets closer to some number. Specifically, the limit of a function \( f(x) \) as \( x \) approaches a point \( c \) is the value \( L \) that \( f(x) \) approaches as \( x \) takes values closer and closer to \( c \).

• If \( \lim_{x \to c} f(x) = L \), it means near \( x = c \), \( f(x) \) gets arbitrarily close to \( L \).
• Limits are foundational for defining continuity and the derivative at a point.

This concept becomes particularly interesting when functions are "patched together," differing at certain isolated points. As seen in the given example, even if two functions are equal almost everywhere, their limits might still diverge at specific locations due to discontinuities, emphasizing the importance of context around a particular point.

Discontinuous Function

A function is said to be discontinuous at a point if there is a break or jump in its output value at that point. Discontinuity plays a critical role in understanding why two seemingly equal functions might not agree in their limits.A common form of discontinuity is jump discontinuity, where a function abruptly changes value:

• Consider a function \( g(x) \) which might be defined to be different from \( f(x) \) at just \( x = 0 \), but otherwise matches \( f(x) \).
• This isolated change can impact the limit as occurred with the function \( g(x) = x \) for \( x eq 0 \) and \( g(0) = 1 \), presenting different limit outcomes than \( f(x) \).

Understanding discontinuities helps in analyzing unexpected behaviors of functions, especially in mathematical proofs and counterexamples.

Counterexample in Mathematics

A counterexample is a single instance or case that disproves a general statement or hypothesis. It is a powerful tool in mathematics to establish the invalidity of universal claims.Let's understand how it is used:

• Suppose we claim "all swans are white," a single black swan is enough to refute this statement.
• In the exercise, the functions \( f(x) = x \) and \( g(x) \) with a differing value at \( x = 0 \) serve as a counterexample because they demonstrate that the same limit does not necessarily follow from equal behavior at all other points.

Using counterexamples can be very simple yet effective in proving or disproving mathematical statements. It teaches us that even claims that seem well-supported by numerous examples can have exceptions, which must be rigorously examined.