Problem 5 - Calculus: Early Transcendentals

Question

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \( \displaystyle \lim_{t \to -2}\frac{t^4 - 2}{2t^2 - 3t + 2} \)

Step-by-Step Solution

Answer

The limit is \( \frac{7}{8} \).

Step 1: Identify the Limit

We evaluate:
Evaluate the limit and justify each step by indicating the appropriate Limit
Law(s).

\( \displaystyle \lim_{t \to -2}\frac{t^4 - 2}{2t^2 - 3t + 2} \)

Step 2: Attempt Direct Substitution

Substitute the approach value. If defined, that is the limit. If indeterminate (\(0/0\)), apply algebraic techniques or L'Hopital's Rule.

Step 3: Apply Limit Laws if Needed

Use sum, product, quotient, and power limit laws to justify the result.

Step 4: Result

The limit is: The limit is \( \frac{7}{8} \).

Key Concepts

Direct Substitution

Direct substitution is a simple and effective technique for evaluating limits, especially when the function is continuous and doesn't result in any undefined forms when evaluated at a specific point.
To apply direct substitution, you replace the variable in the limit expression with the value it's approaching. In our exercise, we directly substitute \( t = -2 \) into \( \frac{t^4 - 2}{2t^2 - 3t + 2} \) and calculate:

• Numerator: \((-2)^4 - 2 = 16 - 2 = 14\)
• Denominator: \(2(-2)^2 - 3(-2) + 2 = 8 + 6 + 2 = 16\)

The result of the division is \( \frac{14}{16} = \frac{7}{8} \).
When direct substitution yields a numerical value, it indicates that the function is continuous at the point and the limit is just that value. Direct substitution is uncomplicated and efficient when applicable, often being the first method to try when evaluating a limit.

Limit Laws

Limit laws are a set of rules that allow us to find limits of functions based on the limits of their parts. Using these rules simplifies complex limit problems by breaking them into simpler pieces.
Some fundamental limit laws include:

• Sum/Difference Law: \( \lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) \)
• 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: \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \), provided \( \lim_{x \to a} g(x) eq 0 \)

In our exercise, the quotient law would typically apply to enable calculation of the overall limit based on the numerator and denominator limits. Since direct substitution directly yields a rational number \( \frac{7}{8} \), the use of these laws confirms that comprehension and simplification have been done correctly. Limit laws are foundational and serve as a backbone for more advanced limit problems and calculus.

Indeterminate Forms

When evaluating limits, one might encounter expressions that initially seem undefined; these are called indeterminate forms. Common types include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), and \( 0\cdot\infty \).
These arise in limits when direct substitution doesn't produce a clear result.
To resolve them, one might need to manipulate the expression further, often using techniques such as factoring, rationalizing, or applying L'Hôpital's rule.
In our presented problem, substituting \( t = -2 \) didn't lead to an indeterminate form, which simplified the solution process. Had it resulted in something like \( \frac{0}{0} \), we would need to use additional algebraic techniques to simplify or transform the expression so that a meaningful limit could be found.
Recognizing indeterminate forms is crucial in deciding how to approach solving a limit, ensuring that misinterpretations and calculation errors are avoided. Understanding and identifying these forms is an essential skill in calculus, paving the way for successful problem-solving and deeper comprehension.