Problem 67 - Calculus: Early Transcendentals

Question

Limits with a parameter Let \(f(x)=\frac{x^{2}-7 x+12}{x-a}\) a. For what values of \(a\), if any, does \(\lim _{x \rightarrow a^{+}} f(x)\) equal a finite number? b. For what values of \(a\), if any, does \(\lim _{x \rightarrow a^{+}} f(x)=\infty ?\) c. For what values of \(a\), if any, does \(\lim _{x \rightarrow a^{+}} f(x)=-\infty ?\)

Step-by-Step Solution

Answer

Question: For the function \(f(x)=\frac{x^{2}-7x+12}{x-a}\), find the values of \(a\) for which the limits exist with a finite value, infinity, or negative infinity as x approaches \(a^{+}\). Answer: The limits exist with a finite value for \(a=3\) and \(a=4\), approach infinity when \(a>4\), and never approach negative infinity.

Step 1: Simplify\(f(x)\)

First, let us factor the numerator and simplify the function as much as possible: \(f(x)=\frac{(x-3)(x-4)}{x-a}\)

Step 2: Identify the discontinuity

Next, we need to identify the points of discontinuity for our function. A rational function like this will have a discontinuity wherever the denominator is equal to zero. Therefore, the discontinuity is at \(x=a\).

Step 3: Determine the limit as \(x\) approaches \(a^{+}\)

Now, we need to determine the limit as \(x\) approaches \(a^{+}\), f(x) = \(\frac{(x-3)(x-4)}{x-a}\). Depending on the value of \(a\), the limit can be finite, infinity, or negative infinity.

Step 4: Consider cases for the limit

We will consider 3 cases regarding the limit: Case a) When the limit equals a finite number: \(\lim_{x\rightarrow a^{+}}\frac{(x-3)(x-4)}{x-a}\) is finite if and only if the numerator also approaches zero. This happens when \(a=3\) or \(a=4\). Case b) When the limit equals infinity: \(\lim_{x\rightarrow a^{+}}\frac{(x-3)(x-4)}{x-a}\) is infinity if the numerator is positive and the denominator approaches zero from the positive side. Let's analyze the sign of \((x-3)(x-4)\) when x is slightly larger than a: If \(a>4\), then \((x-3)(x-4)>0\) and the denominator is also positive. Therefore, the limit equals infinity when \(a>4\). Case c) When the limit equals negative infinity: \(\lim_{x\rightarrow a^{+}}\frac{(x-3)(x-4)}{x-a}\) is negative infinity if the numerator is positive and the denominator approaches zero from the negative side. However, there is no such case to be found since, for \(x>a\), the denominator is always positive. #Conclusion:# a) The limit \(\lim_{x\rightarrow a^{+}}\frac{(x-3)(x-4)}{x-a}\) equals a finite number if \(a=3\) or \(a=4\) b) The limit \(\lim_{x\rightarrow a^{+}}\frac{(x-3)(x-4)}{x-a}\) equals infinity when \(a>4\) c) The limit \(\lim_{x\rightarrow a^{+}}\frac{(x-3)(x-4)}{x-a}\) never equals negative infinity.

Key Concepts

Rational Functions

Rational functions play a crucial role in calculus and are typically expressed as the ratio of two polynomials. For example, the function \(f(x)=\frac{(x-3)(x-4)}{x-a}\) is a rational function. Here, the numerator \((x-3)(x-4)\) and the denominator \(x-a\) are both polynomials. Because rational functions involve a division by a polynomial, understanding the behavior of the denominator is essential. Unlike polynomial functions, rational functions can display complex behaviors such as vertical asymptotes and holes, which affect their limits and continuity.

These functions can be used to model real-world scenarios where relationships between variables involve division. For students, learning about rational functions helps in analyzing how these expressions behave under different conditions, especially when they are combined with limits.

Discontinuity in Functions

In calculus, discontinuity refers to points where a function is not continuous. For rational functions, these points often occur where the denominator is zero. In \(f(x)=\frac{(x-3)(x-4)}{x-a}\), the function becomes discontinuous at \(x=a\), because division by zero is undefined.

Recognizing and dealing with discontinuities is key to mastering calculus. Types of discontinuities include removable and non-removable. A removable discontinuity appears as a 'hole' in the graph, which can often be addressed by simplifying the function, whereas non-removable concerns vertical asymptotes that occur when the function approaches infinity. Understanding discontinuities allows you to predict the behavior of functions in a more detailed manner.

One-Sided Limits

One-sided limits are essential tools in calculus for understanding the behavior of functions as they approach particular points. Unlike standard limits, which consider the behavior as approaches from both left and right, one-sided deals with the approach from either direction alone.

Let's consider the one-sided limit \(\lim_{x\rightarrow a^{+}} f(x)\), which focuses on the approach to \(x=a\) from the positive side. This kind of limit helps us analyze behavior at discontinuities specifically. In our given rational function, knowing whether the numerator becomes zero, or maintains a positive or negative value, allows us to determine if the limit is finite, infinite, or does not exist.

• If the numerator tends to zero when \(x=a\), the limit could be finite, for example, if \(a=3\) or \(a=4\).
• If the numerator is positive and x is slightly larger than a, it might lead to infinity if \(a>4\).
• Negative infinity isn't observed, as analyzed because the denominator never becomes negative as it approaches from the right.

Learning about one-sided limits enhances your ability to tackle discontinuities and infinite values in calculus efficiently.