Problem 13 - Calculus and its applications

Question

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 5}\left(x^{2}-6 x+9\right) $$

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Identify the Function Type

The given function is a polynomial: \( f(x) = x^2 - 6x + 9 \). In this step, we identify that we are dealing with a polynomial function.

Step 2: Know the Limit Laws for Polynomials

According to the Theorem on Limits of Rational Functions, if \( f(x) \) is a polynomial function, \( \lim_{{x \to c}} f(x) = f(c) \). Therefore, for polynomial functions, the limit can be found by direct substitution of \( x = c \).

Step 3: Substitute the Value into the Function

Substitute \( x = 5 \) into the polynomial: \[ f(5) = 5^2 - 6 \times 5 + 9 \]

Step 4: Calculate the Function Value

Compute the simplified result of the previous substitution: \[ 5^2 = 25 \]\[ 6 \times 5 = 30 \]Thus, \[ f(5) = 25 - 30 + 9 \] \[ f(5) = 4 \].

Step 5: Final Step: Conclusion of the Limit

Since \( f(5) = 4 \), therefore, the limit is \[ \lim_{x \to 5} (x^2 - 6x + 9) = 4 \].

Key Concepts

Polynomial Functions

Polynomial functions are expressions composed of variables and constants combined using addition, subtraction, and multiplication. There are no division by a variable in polynomial expressions. Each term in a polynomial is a product of a constant coefficient and variables raised to whole number powers. Common examples include quadratic functions like \( x^2 + 3x + 2 \) and cubic functions like \( x^3 + 2x^2 - x - 1 \).

Key Characteristics of Polynomial Functions:

• Continuous and smooth graphs.
• Defined for all real numbers.
• The degree of the polynomial (the highest power of the variable) tells us the maximum number of roots or solutions the polynomial can have.
• The end behavior of the function is determined by the leading term, the term with the highest power.

The limit of a polynomial function as \( x \) approaches a certain value can be easily found by substituting the value into the polynomial, as demonstrated in the original step-by-step solution.

Rational Functions

Rational functions are the ratio of two polynomial functions, expressed in the form \( \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These functions are defined everywhere except at the values of \( x \) where \( Q(x) = 0 \), called the vertical asymptotes.

Key Properties:

• If the degree of \( P(x) \) (the numerator) and \( Q(x) \) (the denominator) are equal, the horizontal asymptote is the ratio of leading coefficients.
• If the degree of \( P(x) \) is less than the degree of \( Q(x) \), the horizontal asymptote is \( y = 0 \).
• If the degree of \( P(x) \) is greater than the degree of \( Q(x) \), there is no horizontal asymptote.
• Finding limits for rational functions may involve factoring and simplifying before substitution.

Theorem on Limits of Rational Functions helps determine limits by substituting values directly into the simplified rational expression provided \( Q(x) eq 0 \) at that point.

Limit Laws

Limit laws are key rules and properties utilized to simplify and find limits in calculus. They help in understanding how limits interact with different mathematical operations like addition, subtraction, multiplication, and division.

Basic Limit Laws Include:

• Sum Law: \( \lim_{{x \to c}} (f(x) + g(x)) = \lim_{{x \to c}} f(x) + \lim_{{x \to c}} g(x) \)
• Difference Law: \( \lim_{{x \to c}} (f(x) - g(x)) = \lim_{{x \to c}} f(x) - \lim_{{x \to c}} g(x) \)
• Product Law: \( \lim_{{x \to c}} (f(x) \times g(x)) = \lim_{{x \to c}} f(x) \times \lim_{{x \to c}} g(x) \)
• Quotient Law: \( \lim_{{x \to c}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to c}} f(x)}{\lim_{{x \to c}} g(x)} \) provided \( \lim_{{x \to c}} g(x) eq 0 \)
• Constant Multiple Law: \( \lim_{{x \to c}} [k \cdot f(x)] = k \cdot \lim_{{x \to c}} f(x) \)

These laws greatly simplify the process of finding limits, especially when working with complex expressions. When applying these laws, be sure to consider if any simplifying steps are necessary before substituting values to prevent undefined expressions.