Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow 7^{-}} 100$$

Step-by-Step Solution

Answer

The limit is 100.

Step 1: Identify the Expression

The expression provided is for finding the limit of a constant function as it approaches a specific point from the left, which is denoted as \( \lim _{x \rightarrow 7^{-}} 100 \). This means we need to understand what happens to the function \( y = 100 \) as \( x \) gets closer and closer to 7 from the left side.

Step 2: Understand the Behavior of a Constant Function

A constant function, such as \( y = 100 \), has the same value for all values of \( x \). This means that regardless of what \( x \) approaches, the function \( y \) remains at 100. Whether \( x \) approaches 7 from the left, right, or any other direction, the value does not change.

Step 3: Apply Limit Laws for Constant Functions

According to the limit laws for constant functions, the limit of a constant function as \( x \) approaches any value is simply the constant itself. Therefore, \( \lim _{x \rightarrow 7^{-}} 100 = 100 \).

Key Concepts

Constant Function

A constant function is a simple yet important concept in calculus. It is a type of function where, no matter what the input value (or 'x') is, the output value remains the same. This is represented as \( y = c \), where \( c \) is a constant. For example, in our scenario, \( y = 100 \) is a constant function. This means that the output value will always be 100, no matter what the input value of \( x \) might be.
Understanding constant functions can simplify many calculus problems because they are straightforward. Whether you look at this graphically, where the function is a horizontal line, or mathematically, there are no changes or jumps as 'x' changes. This makes calculations involving constant functions much easier to handle and analyze.

Limit Laws

Limit laws provide a set of rules that can be applied to evaluate limits of functions smoothly. For constant functions, there is a specific limit law that states the limit of a constant as the input (\( x \)) approaches any value is the constant itself. This simplifies the evaluation of limits for constant functions drastically.
When we use limit laws, we avoid any complex calculations. For instance, in our exercise, given the function \( y = 100 \), applying the limit law directly tells us that no matter what 'x' is approaching, the limit of \( y \) will be 100. This is because the value of the function does not depend on \( x \) in any way. These laws are crucial as they allow us to break down more complex problems into manageable parts.

Approaching a Point

When we talk about approaching a point in calculus, we're interested in what happens to the values of a function as the input \( x \) gets closer and closer to a certain number. This is a central idea when discussing limits. In our case, \( x \) approaches 7 from the left, indicated by the notation \( x \rightarrow 7^{-} \). This means we observe what the function does as 'x' comes closer to 7, specifically from values less than 7.
This concept is key to understanding limits because it focuses on behavior nearby points rather than the actual point itself. For constant functions, as \( x \) approaches this point, the function remains unchanged, simplifying the discussion. Thus, no matter from which direction \( x \) is approached, or how close it gets to the point, the function value stays the same, illustrating the steadfast nature of limits with constant functions.