Problem 89 - Precalculus with Limits

Question

WRITING Consider the limit of the rational function given by \(p(x) / q(x)\). What conclusion can you make if direct substitution produces each expression? Write a short paragraph explaining your reasoning. (a) \\[\lim_{x\to c} \dfrac{p(x)}{q(x)} = \dfrac{0}{1}\\] (b) \\[\lim_{x\to c} \dfrac{p(x)}{q(x)} = \dfrac{1}{1}\\] (c) \\[\lim_{x\to c} \dfrac{p(x)}{q(x)} = \dfrac{1}{0}\\] (d) \\[\lim_{x\to c} \dfrac{p(x)}{q(x)} = \dfrac{0}{0}\\]

Step-by-Step Solution

Answer

For these types of cases, the results for the limits are; \(0\) for \(\dfrac{0}{1}\), \(1\) for \(\dfrac{1}{1}\), undefined for \(\dfrac{1}{0}\), and indeterminate form requiring further calculation for \(\dfrac{0}{0}\) respectively.

Step 1: Understanding the Meaning of Direct Substitution in Limits

Substituting \(x\) with \(c\) in both \(p(x)\) and \(q(x)\) gives us different fractions each of which has a mathematical and conceptual interpretation. Let's dive into this.

Step 1: Interpretation of \(\lim_{x\to c} \dfrac{p(x)}{q(x)} = \dfrac{0}{1}\)

From the direct substitution, if the limit of the function as \(x\) approaches \(c\) equals \(0\), it means that the function \(p(x)\) approaches \(0\) far more quickly than \(q(x)\) approaches \(1\) as \(x\) nears \(c\). The function therefore approaches \(0\).

Step 2: Interpretation of \(\lim_{x\to c} \dfrac{p(x)}{q(x)} = \dfrac{1}{1}\)

In this case, both \(p(x)\) and \(q(x)\) are approaching \(1\) at the same rate which makes the function's limit equals to \(1\).

Step 3: Interpretation of \(\lim_{x\to c} \dfrac{p(x)}{q(x)} = \dfrac{1}{0}\)

Here, \(p(x)\) approaches \(1\) while \(q(x)\) approaches \(0\) as \(x\) nears \(c\). Mathematically, dividing any non-zero number by \(0\) is undefined, so we can say that this function's limit is undefined as \(x\) approaches \(c\).

Step 4: Interpretation of \(\lim_{x\to c} \dfrac{p(x)}{q(x)} = \dfrac{0}{0}\)

Here both \(p(x)\) and \(q(x)\) are tending to \(0\) as \(x\) approaches \(c\). This case is known as an indeterminate form. It is not possible to determine the value of the limit from this form. Hence, other techniques like L'Hopital's rule or factoring should be used to determine the limit.

Key Concepts

Direct Substitution in Limits

In calculus, direct substitution in limits is often the first method attempted when evaluating the limit of a rational function as it approaches a certain value. To apply direct substitution, one simply plugs the value to which the variable approaches into the function, if this action is possible.

For instance, if you have the limit expression \(\lim_{x\to c} \dfrac{p(x)}{q(x)}\), you would replace the variable \(x\) with the number \(c\) in both the numerator \(p(x)\) and the denominator \(q(x)\). The result can provide immediate insights:

• If the substitution results in a fraction like \(\dfrac{0}{1}\), the function's limit at \(c\) is \(0\).
• If you get a non-zero constant over another non-zero constant, like \(\dfrac{1}{1}\), the limit is the value of that constant.

However, direct substitution is not always successful. When it leads to undefined expressions, additional strategies must be employed to find the limit.

Indeterminate Forms

Indeterminate forms occur in limit problems where the limit of an algebraic expression can't be determined from the expression alone after direct substitution. The most common indeterminate form is \(\dfrac{0}{0}\), but there are others such as \(\infty - \infty\), \(0\cdot\infty\), and \(\dfrac{\infty}{\infty}\).

When direct substitution results in an indeterminate form like \(\dfrac{0}{0}\), this signals the need for additional techniques to determine what happens to the function near the point of interest. It might be that as the variable approaches a specific value, both the numerator and denominator approach zero, but at different rates. In such cases, the limit could exist and could be any finite number or even infinity.

L'Hopital's Rule

L'Hopital's rule is a powerful tool for finding limits that result in indeterminate forms. When evaluating \(\lim_{x\to c} \dfrac{p(x)}{q(x)}\) leads to \(\dfrac{0}{0}\) or \(\dfrac{\infty}{\infty}\), L'Hopital's rule can often provide a way forward. This rule states that under certain conditions, the original limit can be computed by taking the derivative of the numerator and denominator separately and then evaluating the limit of this new rational function.

For example, if direct substitution in the limit \(\lim_{x\to c} \dfrac{p(x)}{q(x)}\) yields \(\dfrac{0}{0}\), then, according to L'Hopital's rule, \(\lim_{x\to c} \dfrac{p(x)}{q(x)} = \lim_{x\to c} \dfrac{p'(x)}{q'(x)}\), provided that the derivatives exist and the limit on the right side is determinate. By repeating this process if necessary, one can often resolve the indeterminate form and find the limit.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as its argument gets arbitrarily close to a certain point. Precisely, the limit of a function \(f(x)\) as \(x\) approaches a value \(c\) is the value that \(f(x)\) approaches as \(x\) comes arbitrarily close to \(c\).

It's important to understand that the limit is about the value the function is approaching, not necessarily the value the function actually reaches. This nuance allows us to discuss the behavior of functions at points where they might be undefined or discontinuous. When computing limits, one may use a variety of approaches - direct substitution being the simplest when it works, and techniques like factoring, simplifying, or L'Hopital's rule when more complex cases arise.