Problem 43 - Calculus for Biology and Medicine

Question

In Problems \(37-54\), 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 23.5.

Step 1: Direct Substitution

First, attempt to evaluate the limit by directly substituting the value of \(x\) approaching -3. When doing so, substitute \(-3\) into the function: \( \frac{(-3)^3 - 20}{-3 + 1} = \frac{-27 - 20}{-2} = \frac{-47}{-2} \). However, since our primary approach was algebraic, let's proceed to check for simplification or re-evaluation to verify.

Step 2: Evaluate Simplification

There's no need for any factorization or simplification beyond direct substitution in this specific case, as the expression does not zero out at either the numerator or denominator just by substituting \(x = -3\). Since \( (-3)^3 - 20 = -47 \) and \( -3 + 1 = -2 \) aren't indeterminate forms, direct substitution worked here.

Key Concepts

Direct Substitution

Direct substitution is one of the simplest methods to evaluate limits. The idea is to plug the given value of the variable into the function directly and compute the result. This technique is very convenient when the function is continuous at the point of interest. For example, in the problem given, we found the limit of the function \( \lim _{x \rightarrow-3} \frac{x^{3}-20}{x+1} \) by substituting \( x = -3 \) directly into the expression. This gave us: - The numerator becomes \( (-3)^3 - 20 \), which simplifies to \( -47 \).
- The denominator becomes \( -3 + 1 \), simplifying to \( -2 \).
This results in \( \frac{-47}{-2} = 23.5 \). By directly substituting, we quickly find that the limit is \( 23.5 \). Thus, if a function does not become undefined or lead to an indeterminate form like \( \frac{0}{0} \), direct substitution provides an effortless solution.

Limit Laws

Limit laws are essential tools for limit evaluation, especially when direct substitution doesn't work, such as becoming an indeterminate form. These laws provide rules to simplify complex limits into manageable problems. Some fundamental limit laws include: - **Sum Law:** \( \lim_{x\to a} [f(x) + g(x)] = \lim_{x\to a} f(x) + \lim_{x\to a} g(x) \)
- **Difference Law:** \( \lim_{x\to a} [f(x) - g(x)] = \lim_{x\to a} f(x) - \lim_{x\to a} g(x) \)
- **Product Law:** \( \lim_{x\to a} [f(x) \cdot g(x)] = \lim_{x\to a} f(x) \cdot \lim_{x\to a} g(x) \)
- **Quotient Law:** \( \lim_{x\to a} \frac{f(x)}{g(x)} = \frac{\lim_{x\to a} f(x)}{\lim_{x\to a} g(x)} \) (where \( \lim_{x\to a} g(x) eq 0 \))
These laws are steps toward simplifying expressions before applying direct substitution and can reveal whether more complex techniques, like factoring or rationalizing, are needed.

Algebraic Simplification

Algebraic simplification often comes into play when direct substitution initially results in an undefined expression or an indeterminate form. By rewriting or simplifying the function, you can resolve these issues to apply limit laws effectively. Here are some common simplification strategies: - **Factoring:** Useful when the function shows common factors, especially in polynomials.
- **Rationalizing:** Involves multiplying by a conjugate if dealing with roots.
- **Canceling Common Terms:** Helps in removing indeterminate forms by simplifying fractions.
For the given problem, algebraic simplification wasn't necessary. Direct substitution immediately provided a defined value, meaning the limit could be determined straight away without manipulation. Nonetheless, knowing simplification techniques is crucial for dealing with more complex problems where initial substitution fails to yield a straightforward result.