Problem 15 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow-4}(x+3)^{2} $$

Step-by-Step Solution

Answer

The limit as \(x\) approaches -4 of \((x+3)^{2}\) is 1.

Step 1: Identify the Function and the Limiting Value

The function given is \((x+3)^2\), and the value of \(x\) which we are finding the limit at is -4.

Step 2: Substitute the Limiting Value into the Function

Substitute \(x = -4\) into the function \((x+3)^2\). Hence, the equation becomes \((-4+3)^2\).

Step 3: Simplify and Solve

\((-4+3)^2 = (-1)^2 = 1\), as the square of -1 gives 1.

Key Concepts

Limits

In calculus, limits help us understand the behavior of functions as they approach a particular point. They tell us what a function is approaching, even if it doesn't ever reach that point. This is crucial in calculus because it enables us to analyze and predict the behavior of functions that might be undefined or difficult to evaluate at certain points.Limits are the foundation of derivatives and integrals. Without them, calculus would not be possible. For instance, solving the problem, \[\lim_{x \rightarrow -4} (x+3)^2\], involves determining what the function \((x+3)^2\) becomes as \(x\) gets very close to \(-4\). By finding limits, we can make sense of situations where substituting a number directly into a function provides undefined or non-intuitive results.

Functions

Functions are mathematical entities that assign a unique output value (or a set of values) to every input value. They often appear as expressions, such as \((x+3)^2\), where \(x\) is the variable input and \((x+3)^2\) is the expression that determines the output value.We deal with functions frequently when working with limits. Understanding functions includes knowing their domain (all possible input values) and range (all possible output values). In the given problem, the function \((x+3)^2\) accepts any real number as an input, meaning its domain is the entire set of real numbers. Evaluating limits, however, may require more than just substituting numbers, as it involves analyzing the behavior of that function around a particular input point.

Limit Substitution

Limit substitution is a straightforward method used whenever the limit can be found by directly plugging the limit point into the function. This method assumes that the function is continuous at the point, meaning no breaks or jumps in the graph occur at that value.In the problem we discussed, you can directly substitute \(x = -4\) into the function \((x+3)^2\) because this function is continuous everywhere, particularly at \(-4\).

• Step 1: Identify the function and the limit point, in this case, \(x = -4\).
• Step 2: Substitute \(-4\) into the function \((x+3)^2\).

This results in calculating \((-4+3)^2 = (-1)^2\), which simplifies to 1. Therefore, the limit is 1.

Evaluating Limits

Evaluating limits involves analyzing the result of a function as the input approaches a certain value. It is crucial when dealing with irregularities or asymptotic behavior in functions.There are several methods for evaluating limits:

• Direct Substitution: Directly substitute the limit value into the function if it results in a meaningful number, as calculated before with \((-4+3)^2 = 1\).
• Factoring: Simplify the function to cancel any problematic terms if a direct substitution results in an indeterminate form like \(\frac{0}{0}\).

Understanding the function's behavior near the limit value is essential for more complex functions. Sometimes, evaluating limits might involve using techniques like L'Hôpital's rule or trigonometric identities, particularly for more complicated cases.