Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated limit. \(\lim _{x \rightarrow-2}-3\)

Step-by-Step Solution

Answer

The limit of a constant function is always the constant itself. Therefore, the indicated limit is: \(\lim _{x \rightarrow -2}-3 = -3\)

Step 1: Identify the type of function

In this case, we are given a constant function, -3. This means that the function does not depend on the variable x.

Step 2: Recall the property of limit for constant functions

The property of the limit for constant functions states that the limit of a constant function as x approaches any value is the constant itself. In other words, \(\lim _{x \rightarrow a}C = C\), where C is a constant and a is any real number.

Step 3: Apply the limit to the constant function

Now, let's apply the property of the limit to our given function: \[\lim _{x \rightarrow -2}-3\] According to the property we mentioned, the limit should be equal to the constant function itself, i.e., -3.

Step 4: Conclusion

So, the indicated limit is: \[\lim _{x \rightarrow -2}-3 = -3\]

Key Concepts

Constant Function

A constant function is one of the simplest types of functions in calculus. It is defined as a function that gives the same output value, regardless of the input. In mathematical terms, for a constant function, if you have a function \( f(x) = C \), the output will always be \( C \), no matter what value \( x \) takes.
For example, in the function we are considering, \( f(x) = -3 \), the output will always be -3, irrespective of \( x \).

• The graph of a constant function is a horizontal line.
• The slope of this line is zero because the function does not change.
• Constant functions are straightforward because they involve no variables in the calculation of their limits.

Understanding constant functions helps simplify limit evaluation since their limits are predictable based on the property described below.

Properties of Limits

The properties of limits in calculus allow us to make conclusions about the limits of functions based on their behaviors or characteristics. An essential property relates to constant functions. The limit of a constant function is simply the constant value itself, no matter where the limit is taken from.
This property can be expressed as:

• If \( f(x) = C \) is a constant function, then \( \lim_{x \to a} f(x) = C \) for any real number \( a \).
• This simplifies many calculations, as seen in the example \( \lim_{x \to -2} (-3) = -3 \).

Apart from constant functions, properties of limits help us calculate complex limits. They function as foundational rules for evaluating limits, making complex calculus problems more approachable.

Calculus Basics

Getting a grasp on the basics of calculus is crucial for tackling more advanced concepts like derivatives and integrals. Calculus deals with change and motion, primarily through the study of limits, derivatives, and integrals.

• Limits are fundamental — they investigate what happens to functions as they approach a particular point, even if they never actually reach that point. This is crucial for understanding continuous functions and the rates at which things change.
• The concept of a limit lays the groundwork for derivatives, which measure how a function changes at any point, and integrals, which measure the total accumulation of quantities.

By mastering the concept of limits, including for constant functions, students build a strong base for exploring more complex topics in calculus, crucial for their mathematical journey.