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. If the limit of \(f(x)\) as \(x\) approaches \(c\) is \(0,\) then there must exist a number \(k\) such that \(f(k)<0.001\)

Short answer

The statement is false. While it's generally possible to find a 'k' for which \(f(k)<0.001\) when the limit of \(f(x)\) as \(x\) approaches a certain value is 0, it is not universally true for all functions. A constant function staying above 0.001 provides a counterexample.

Step-by-step solution

Understand the Statement

The statement is saying that if the limit of a certain function \(f(x)\) as \(x\) tends towards 'c' is zero, then there must exist a certain 'k' where \(f(k)\) will be less than 0.001.

Analyze the Statement

Let's consider the function \(f(x) = x^2\). The limit of this function as \(x\) approaches 0 is indeed 0, since \(\lim_{x \to 0}x^2=0\). Now, according to the statement, there should exist a 'k' such that when plugged in the function \(f(x)\), the output is less than 0.001. For our function, we can generally find such a 'k' within the vicinity of 0, since as we move close to 0, \(x^2\) will give a very small output.

Evaluate the Statement

In the previous step, we saw that there exist numbers 'k' that will give \(f(k)<0.001\), but this is not necessarily true if our function \(f(x)\) never takes values below 0.001. For instance, consider the function \(f(x) =0.002\). As \(x\) approaches any value, \(f(x)\) stays constant at 0.002, thus its limit as \(x\) approaches any value is 0.002. But it's not possible to find a 'k' such that \(f(k)<0.001\). Therefore, the statement is false.

Key concepts

Limit of a Function

Understanding the limit of a function is a cornerstone of calculus. It refers to the behavior of a function as its input approaches a certain point. The formal definition can be a bit intimidating at first, but at its core, it is quite intuitive. When we say, for example, that the limit of a function \( f(x) \) as \( x \) approaches \( c \) is \( L \), we’re describing the value that \( f(x) \) gets closer and closer to as \( x \) gets near \( c \).

However, it's crucial to understand that the limit does not necessarily equal the function's value at \( c \) itself; it's about the trend as we approach \( c \), not the destination. For instance, even if \( f(c) \) is undefined or differs from \( L \), the limit as \( x \) approaches \( c \) can still be \( L \).

To improve comprehension, consider visualizing this on a graph—imagine drawing a line closer and closer to a point on the function's curve. The height of the curve just before reaching the point is the visual representation of the limit. It's the 'almost there' value, regardless of whether the curve actually makes it to that value at the point itself.

Approaching a Point

The phrase approaching a point in calculus is tied closely to the concept of limits. It means that as the input value of a function gets closer to a certain number, we are interested in understanding what happens to the output value. When we say that \( x \) is approaching \( c \), we are not concerned with what happens exactly at \( c \), but rather how the function behaves as we get arbitrarily close to \( c \) from either direction.

Imagine walking along a path and getting ever nearer to a landmark without actually touching it—that is what we mean by approaching in a mathematical sense. In calculus, this concept helps us solve problems involving continuous functions, where the input can get infinitesimally close to any given point, providing us valuable insight into the function's behavior around that point.

This idea is pivotal when dealing with real-world scenarios that don't have exact values, allowing us to predict what happens as we near certain conditions, such as time approaching the end of a race, or an item's falling velocity as it gets closer to the ground.

Counterexample in Calculus

In the realm of mathematics, and especially in calculus, a counterexample is a powerful tool used to disprove general statements or findings. It is essentially an example that demonstrates that a statement is not always true, no matter how intuitive or self-evident it may seem. When a statement claims that something is true for all cases, we can disprove it by finding just one case where the statement fails—this single occurrence is the counterexample.

In the exercise we're considering, finding a function where the limit does not lead to a value below a certain threshold, yet approaches a specific value, serves as a counterexample to the original claim. Counterexamples help sharpen our understanding by setting boundaries for the applicability of theoretical assertions.

In teaching, providing counterexamples can be an excellent way to encourage critical thinking. Students learn to question and test claims rigorously, rather than accepting them at face value. This process fosters deeper comprehension and equips students with a pragmatic approach to problem-solving in calculus, where recognizing the limits (pun intended) of a statement's validity is as important as understanding the statement itself.