Problem 82 - Calculus and its applications

Question

Let \(f\) and \(g\) be differentiable over an open interval containing \(x=a\). If $$ \lim _{x \rightarrow a}\left(\frac{f(x)}{g(x)}\right)=\frac{0}{0} \quad \text { or } \quad \lim _{x \rightarrow a}\left(\frac{f(x)}{g(x)}\right)=\frac{\pm \infty}{\pm \infty} $$ and if \(\lim _{x \rightarrow a}\left(\frac{f^{\prime}(x)}{g^{\prime}(x)}\right)\) exists, then $$ \lim _{x \rightarrow a}\left(\frac{f(x)}{g(x)}\right)=\lim _{x \rightarrow a}\left(\frac{f^{\prime}(x)}{g^{\prime}(x)}\right) $$ The forms \(0 / 0\) and \(\pm \infty / \pm \infty\) are said to be indeterminate. In such cases, the limit may exist, and l'Hôpital's Rule offers a way to find the limit using differentiation. For example, in Example 1 of Section \(1.1,\) we showed that $$ \lim _{x \rightarrow 1}\left(\frac{x^{2}-1}{x-1}\right)=2 $$ Since, for \(x=1,\) we have \(\left(x^{2}-1\right)(x-1)=0 / 0,\) we differentiate the numerator and denominator separately, and reevaluate the limit: $$ \lim _{x \rightarrow 1}\left(\frac{x^{2}-1}{x-1}\right)=\lim _{x \rightarrow 1}\left(\frac{2 x}{1}\right)=2 $$ Use this method to find the following limits. Be sure to check that the initial substitution results in an indeterminate form. $$ \lim _{x \rightarrow 10}\left(\frac{x^{2}+x-110}{x-10}\right) $$

Step-by-Step Solution

Answer

The limit is 21.

Step 1: Check for Indeterminate Form

Substitute \( x = 10 \) into the equation \( \frac{x^2 + x - 110}{x - 10} \). We have: \( \frac{10^2 + 10 - 110}{10 - 10} \) which simplifies to \( \frac{100 + 10 - 110}{0} \) or \( \frac{0}{0} \). This confirms the form is indeterminate.

Step 2: Apply l'Hôpital's Rule

Since the limit is in the indeterminate form \( \frac{0}{0} \), we can apply l'Hôpital's Rule. Differentiate the numerator and the denominator separately. The derivative of the numerator, \( x^2 + x - 110 \), is \( 2x + 1 \). The derivative of the denominator, \( x - 10 \), is \( 1 \).

Step 3: Evaluate the Limit of the Derivatives

Reevaluate the limit using the derivatives obtained: \( \lim_{x \rightarrow 10} \left( \frac{2x + 1}{1} \right) = \lim_{x \rightarrow 10} (2x + 1) \). Substitute \( x = 10 \), giving \( 2(10) + 1 = 20 + 1 = 21 \).

Step 4: Conclusion

The limit is \( 21 \). Therefore, the given problem \( \lim_{x \rightarrow 10} \left( \frac{x^2 + x - 110}{x - 10} \right) \) evaluates to \( 21 \) using l'Hôpital's Rule.

Key Concepts

Indeterminate Forms

An indeterminate form arises when the precise value of a limit is unclear, usually because the value could be tending towards more than one possible number. In the context of limits, indeterminate forms are expressions like \(\frac{0}{0}\) or \(\frac{\pm \infty}{\pm \infty}\). Unlike other forms that have a definitive answer, indeterminate forms need further analysis to determine their actual limit value. When substituting a value into a function results in an indeterminate form, this signals that direct evaluation of the limit won’t provide a complete picture of the function's behavior at the point.
When faced with an indeterminate form, one approach to resolve it is to manipulate the expression algebraically, such as factoring or expanding, to eliminate terms that lead to zero in the denominator. Once simplified, you can often directly compute the limit. However, when these algebraic techniques don't work, and especially in cases involving calculus, applying differentiation techniques like l'Hôpital's Rule may be necessary.

Differentiation

Differentiation is a powerful mathematical tool in calculus used to find the rate of change of a function. More formally, the derivative of a function at a point provides the slope of the tangent line to the function’s graph at that point.

To differentiate a function, we apply derivative rules, such as power, product, and chain rules, among others. Each rule guides how to handle different types of functions, be they polynomials, exponentials, or others. The result of differentiation is another function showing how the original function changes.
In the context of l'Hôpital's Rule, differentiation allows us to transform an indeterminate form into a simpler form in which the limit can be directly evaluated. For example, when facing an expression like \( \frac{f(x)}{g(x)} \), where direct substitution results in an indeterminate form, both the numerator and denominator are differentiated separately. This reduces the complexity of the original fraction, often resolving the indeterminate form into a determinate one that can be easily evaluated.

Calculus Limits

Limits are a foundational concept in calculus, critical for understanding the behavior of functions as they approach specific points or infinity. The limit of a function at a particular point refers to the value that the function's output gets closer to as the input approaches that point.

Calculus limits are crucial for defining derivatives and integrals. A firm understanding of limits enables deeper insights into the continuity and differentiability of functions.

• Evaluating Limits: To evaluate a limit, you can often substitute the point into the function and solve directly. However, if this results in an indeterminate form, additional techniques like algebraic simplification or l'Hôpital's Rule are necessary.
• l'Hôpital's Rule: This rule applies when limits result in indeterminate forms \(\frac{0}{0}\) or \(\frac{\pm\infty}{\pm\infty}\). It involves differentiating the numerator and the denominator separately and retrying the limit.

By mastering the art of determining limits, you will find that many problems in calculus become more straightforward, as limits form the basis upon which other operations are built. Consequently, they are invaluable tools in approximating quantities and solving real-world problems.