Problem 39 - Calculus for Biology and Medicine

Question

In Problems \(37-54\), use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-5}\left(4+2 x^{2}\right) $$

Step-by-Step Solution

Answer

The limit is 54.

Step 1: Identify the Direct Substitution

The first step in evaluating a limit is to directly substitute the value that the variable approaches into the expression. In this case, substitute \( x = -5 \) into the expression \( 4 + 2x^2 \).

Step 2: Substitute the Value

Substitute \( x = -5 \) into the expression \( 4 + 2(-5)^2 \), giving you \( 4 + 2(25) \).

Step 3: Simplify the Expression

Now simplify the expression from the substitution step: \( 2(25) = 50 \), then \( 4 + 50 = 54 \).

Step 4: Write the Final Limit Solution

After simplifying, you find that \[ \lim _{x \rightarrow-5}\left(4+2 x^{2}\right) = 54 \]. This is the final value of the limit.

Key Concepts

Limit Laws

Limit laws simplify the process of evaluating limits by giving guidelines to follow when dealing with different mathematical operations. They allow us to break down complicated expressions into easier parts, making the evaluation process more manageable. These laws help us:

• Preserve addition and subtraction: The limit of a sum or difference is the sum or difference of the limits.
• Multiply and divide: The limit of a product is the product of the limits, and the limit of a quotient is the quotient of their limits provided the denominator's limit is not zero.
• Apply powers and roots: The limit of a power is the power of the limit, and this works similarly for roots.

In the provided exercise, we use these laws to simplify our limit evaluation by substituting and computing each operation step-by-step.

Direct Substitution

Direct substitution is a straightforward method of evaluating limits by plugging the approaching value of the variable directly into the expression. This method works well when dealing with expressions that are continuous at the point of interest.
For the given example, since the expression is a polynomial, which is continuous everywhere, we can simply substitute \( x = -5 \) directly:

• Take the given function \( 4 + 2x^2 \).
• Substitute \( x = -5 \) into the function: \( 4 + 2(-5)^2 \).
• Simplify to find the limit.

This technique is helpful because it bypasses the need for more complex methods, and directly leads us to the final answer.

Simplification

Simplification is crucial when evaluating limits to make expressions easier to solve. Once you substitute the approaching value into the function, you need to simplify the expression step-by-step:

• Calculate any power or product inside the expression first, such as \( 2(-5)^2 = 2 \times 25 \).
• Perform basic arithmetic: \( 2 \times 25 = 50 \), leading to \( 4 + 50 = 54 \).

Simplifying ensures that you avoid mistakes and reach the most reduced form of the expression, which allows for an easier evaluation of the limit outcome.

Calculus Steps

The process of evaluating limits involves systematic steps. Calculus teaches us these tools to better understand functions as variables approach specific values. Here's a recap of the steps used in the problem:

• **Step 1:** Identify if direct substitution can be used. For polynomials, it often will be.
• **Step 2:** Substitute the relevant value into the function, as \( x = -5 \) was substituted here.
• **Step 3:** Simplify the result of the substitution to calculate the limit.
• **Step 4:** State the final value of the limit you found from simplification.

These steps guide students in methodically addressing limit problems, ensuring clarity and accuracy in calculus solutions.