Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit, if it exists. $$\lim _{x \rightarrow 6}(-5)$$

Step-by-Step Solution

Answer

The limit is -5.

Step 1: Understand the Concept

The expression given is a constant function, where the function is always \(-5\) regardless of the value of \(x\). In limits, a constant function does not change as \(x\) approaches any number.

Step 2: Evaluate the Limit

Since \(-5\) is a constant function, the limit of a constant as \(x\) approaches any value is simply the constant itself. Therefore, the limit of \(-5\) as \(x\) approaches 6 is \(-5\).

Key Concepts

Constant Function

A constant function is one of the simplest types of functions in mathematics. It is a function that remains the same for any input value. For example, if you have a constant function like

• \(f(x) = -5\)

It means that \(f(x)\) will always be

• \(-5\)

no matter what the value of \(x\) is. This property simplifies the process of finding limits,because the output doesn’t change when the input changes. In other words, the function's graph is a horizontal line on the coordinate plane. This is crucial when evaluating limits, as we'll see next.

Evaluate Limits

Evaluating limits can sometimes be tricky, but constant functions make it much easier. The limit of a function as \(x\) approaches a particular value describes the value that \(f(x)\) gets closer to, as \(x\) gets closer to a certain point. However, with constant functions, you don't have to worry about complicated calculations.

• For any constant function, like \(f(x) = -5\), the limit as \(x\) approaches any number is always just the constant itself.
• This is because the value of the function doesn't change; it's always consistent.

So, if you are asked to find \(\lim_{x ightarrow 6}(-5)\), the answer is straightforward: it’s simply

• \(-5\)

This principle applies to all constant functions, making them predictable and easy to handle.

Approaching a Value

When discussing limits, a key concept is the idea of 'approaching a value'. This involves considering what happens to the function's output as the input gets closer to a specific number. For many functions, as \(x\) approaches a number, the output value of the function gets arbitrarily close to some limit.

But with a constant function the situation is much simpler. As \(x\) gets closer to any value, the function's value remains unchanged. For instance,

• for \(f(x) = -5\), as \(x\) approaches 6 (or any other number), \(f(x)\) continues to remain \(-5\)

This constancy makes evaluating the limit stress-free, as you do not have to worry about the specific behavior of the function as it nears any given value; it simply stays the same. This fundamental aspect makes understanding the behavior of constant functions under limits truly straightforward.