Question

Given \(y=f(x)=\frac{x^{2}-9 x+20}{x-4}\) (a) Is it possible to apply the quotient limit theorem to find the limit of this function as \(x \rightarrow 4 ?\) (b) is this function continuous at \(x=4 ?\) Why? (c) Find a function which, for \(x \neq 4,\) is equivalent to the given function, and obtain from the equivalent function the limit of \(y\) as \(x \rightarrow 4\)

Short answer

(a) No, direct application is not possible due to zero denominator at \( x=4 \). (b) Not continuous at \( x=4 \), function value undefined there. (c) Limit is \(-1\).

Step-by-step solution

Understand the function

First, observe that the function is given by \( y = f(x) = \frac{x^2 - 9x + 20}{x-4} \). Notice that the function is a rational function consisting of a polynomial in the numerator and a polynomial in the denominator.

Evaluate applying quotient limit theorem (Part a)

To apply the quotient limit theorem, the denominator must be non-zero as \( x \rightarrow 4 \). However, substituting \( x = 4 \) in the denominator of \( \frac{x^2 - 9x + 20}{x-4} \) results in division by zero, since the denominator is zero. Therefore, it is not possible to directly apply the quotient limit theorem in this case.

Check continuity at x = 4 (Part b)

For continuity at \( x = 4 \), the function must be defined at \( x = 4 \) and the limit as \( x \) approaches 4 must equal the function value at 4. However, the function is not defined at \( x = 4 \) because the denominator becomes zero, resulting in an undefined value. Hence, the function is not continuous at \( x = 4 \).

Simplify the function (Part c)

Simplify the expression \( \frac{x^2 - 9x + 20}{x-4} \) by factoring the numerator: \( x^2 - 9x + 20 = (x-4)(x-5) \). The function simplifies to \( y = \frac{(x-4)(x-5)}{x-4} \) for \( x eq 4 \). This can be further simplified by canceling the \( x-4 \) factor: \( y = x-5 \). For \( x eq 4 \), \( y = x-5 \) is equivalent to the given function.

Find the limit using the simplified function (Part c)

Use the simplified expression \( y = x-5 \) to find \( \lim_{x \to 4} y \). Substitute \( x = 4 \) into \( y = x-5 \) to get \( y = 4-5 = -1 \). Therefore, the limit of \( y \) as \( x \rightarrow 4 \) is \(-1\).

Key concepts

Continuity

In calculus, continuity is a fundamental concept that helps us understand how functions behave. A function is considered continuous at a point if it satisfies three conditions:
1. The function is defined at the point.
2. The limit of the function exists as it approaches the point.
3. The limit value and the function value at this point are the same.
If any of these conditions are not met, the function is not continuous at that point.
In the given exercise, we consider the function \(f(x)=\frac{x^{2}-9x+20}{x-4}\). If you plug \(x = 4\) directly into the function, you encounter a problem: the expression involves division by zero. Thus, the function is not defined at \(x=4\). Without a defined value, continuity cannot be established at this point. Therefore, the function is discontinuous at \(x=4\). Understanding this idea helps simplify further problem-solving tasks, such as determining limits and finding equivalent functions.

Rational Functions

Rational functions are ratios of two polynomials. They are represented in the form \( \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomial expressions, and \(Q(x)\) is not zero. The function \(f(x)=\frac{x^{2}-9x+20}{x-4}\) is a classic example of a rational function.
These functions are defined for all real numbers except when the denominator is zero.

• A key feature of rational functions is that they can reveal discontinuities, often leading to undefined points, like with \(x=4\) in this example, as the denominator becomes zero.
• Such points can often be resolved via simplification.

Rational functions can typically be simplified by factoring both the numerator and the denominator and canceling common factors. This often removes point discontinuities, making computation of limits clearer, as demonstrated in the exercise where \(f(x)\) simplifies to \(x-5\) for \(x eq 4\). This technique is essential to find simpler expressions that are easier to work with, especially when analyzing limits and continuity.

Limits

The concept of limits in calculus is essential for understanding the behavior of functions as they approach a particular value. The limit of a function \(f(x)\) as \(x\) approaches a number \(a\) is the value that \(f(x)\) gets closer to as \(x\) gets closer to \(a\).

• Limits are vital for defining derivatives and integrals.
• They help in understanding the continuity of functions.

In the exercise provided, directly substituting \(x = 4\) in the function \(\frac{x^{2}-9x+20}{x-4}\) gives an undefined form \( \frac{0}{0} \). However, by simplifying the function to \( y = x-5 \) for \( x eq 4 \), calculating the limit becomes straightforward. You simply evaluate \(\lim_{x \to 4} (x-5)\), resulting in \(-1\). This limit describes the behavior of the function as \(x\) approaches 4, even if the function itself is not defined at 4, highlighting the importance of limits in capturing the essence of functions near discontinuities.

Quotient Limit Theorem

The Quotient Limit Theorem is a principle in calculus that is used to determine the limit of the quotient of two functions. It states that if \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), and \(M eq 0\), then \(lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}\). In other words, the limit of a quotient is the quotient of the limits, provided the denominator of the quotient does not approach zero.

In the context of the exercise, since the denominator \(x-4\) yields zero as \(x\) approaches 4, the Quotient Limit Theorem cannot be directly applied, resulting in an undefined form.
Identifying whether the theorem is applicable is crucial in analyzing limits of rational functions accurately.
Sometimes, evaluating or simplifying the function near the point of concern can clarify the limit, aiding in bypassing the initial hurdle of undefined zero denominator situations. By simplifying the function, continuity and easier limit calculations are achieved, as in the transition from \(\frac{x^{2}-9x+20}{x-4}\) to \(x-5\).