Problem 45 - Calculus for Biology and Medicine

Question

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-3} \frac{x^{3}-20}{x+1} $$

Step-by-Step Solution

Answer

The limit is \( \frac{47}{2} \).

Step 1: Understand the question

The question asks: In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-3} \frac{x^{3}-20}{x+1} $$

Step 2: Apply Biology knowledge

Using fundamental Biology concepts to analyze the problem.

Step 3: Formulate the answer

The limit is \( \frac{47}{2} \).

Step 4: Conclude

The answer is: The limit is \( \frac{47}{2} \).

Key Concepts

Limit Laws

Limit laws in calculus are a set of rules that help us evaluate limits of functions systematically. When dealing with limits, these laws are essential tools. There are a few key limit laws that you should be familiar with:

• Sum Law: The limit of a sum is the sum of the limits.
• Difference Law: The limit of a difference is the difference of the limits.
• Product Law: The limit of a product is the product of the limits.
• Quotient Law: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.
• Constant Multiple Law: The limit of a constant times a function is the constant times the limit of the function.

By applying these laws correctly, we can break down complex expressions and evaluate them in chunks. In the exercise provided, limit laws were briefly mentioned but not directly used because direct substitution was possible immediately. However, understanding these laws helps us confirm the correctness of direct evaluations and handle more complex situations.

Direct Substitution in Calculus

Direct substitution is one of the simplest methods for evaluating limits. When a function is continuous at a point, you can directly substitute the point into the function to find the limit. It is a straightforward approach that requires minimal computation.
For the function in the exercise:\[\lim_{x \rightarrow -3} \frac{x^{3}-20}{x+1}\]You simply substitute \(-3\) for \(x\) directly:

• Calculate the numerator: \((-3)^3 - 20 = -47\)
• Calculate the denominator: \(-3 + 1 = -2\)
• Evaluate the expression: \(\frac{-47}{-2} = \frac{47}{2}\)

Since this direct approach did not yield an undefined expression (e.g., division by zero), the limit was successfully evaluated as \(\frac{47}{2}\). Direct substitution greatly simplifies the process, especially when polynomials and simple rational functions are involved, and when no indeterminate forms like \(\frac{0}{0}\) occur.

Evaluating Limits

Evaluating limits is a fundamental skill in calculus that involves determining the value that a function approaches as the input approaches a certain point. Usually, we start by trying direct substitution, as it is the quickest way to find the limit if the function is continuous and well-defined at that point.
If direct substitution leads to an indeterminate form, alternative methods like factoring, conjugates, or special techniques might be necessary. Seeing that in our exercise, the function was suitable for direct substitution, it indicates that the function is continuous at \( x = -3 \). This keeps the evaluation simple.
Following this initial check, we rely on limit laws if simplification is necessary. For more involved functions, applying the appropriate limit laws or algebraic manipulation ensures accuracy. Overall, developing proficiency in evaluating limits requires practice and familiarity with these fundamental methods, ensuring a strong foundation in calculus.