Problem 8 - Technical Mathematics with Calculus

Question

Evaluate each limit. $$\lim _{x \rightarrow 5} \frac{x^{2}-12 x+35}{5-x}$$

Step-by-Step Solution

Answer

The limit does not exist.

Step 1: Factorize the numerator

First, factorize the quadratic expression in the numerator. The quadratic can be written as \(x^{2}-12x+35 = (x-7)(x-5)\).

Step 2: Simplify the fraction

Next, simplify the fraction by canceling the \(x-5\) term in the numerator with the \(5-x\) term in the denominator, noting that \(5-x\) is \(x-5\) multiplied by -1, which gives us \(\frac{(x-7)(x-5)}{5-x} = \frac{(x-7)}{-(x-5)} = \frac{-(x-7)}{x-5}\).

Step 3: Evaluate the limit

Finally, evaluate the limit by substituting \(x=5\) into the simplified expression: \(\lim _{x \rightarrow 5} \frac{-(x-7)}{x-5} = \frac{-(5-7)}{5-5} = \frac{-(-2)}{0}\). Since the denominator approaches 0, this is a case of an indeterminate form. We can conclude that the limit does not exist because the function approaches negative infinity as x approaches 5 from the left and positive infinity as x approaches 5 from the right.

Key Concepts

Limit of a Function

In mathematics, when we talk about the limit of a function, we are referring to the behavior of that function as the input values approach a certain point. It's a fundamental concept in calculus used to understand the tendency of functions as variables get incredibly close to a particular value. For example, when asked to evaluate \(\lim _{x \rightarrow a} f(x)\), we're seeking the value that \(f(x)\) approaches as \(x\) gets closer and closer to \(a\).

However, it's important to note that the limit does not always equal the function's value at that point — in fact, sometimes the function might not even be defined at \(a\)! The limit is more about the journey of the \(x\) values towards the number \(a\), rather than the destination itself. In the case of our exercise, we're interested in what happens as \(x\) gets infinitely close to 5, not necessarily what occurs when \(x\) is exactly 5. This distinction is crucial for understanding how limits describe the behavior of functions around points of interest.

Factoring Polynomials

Factoring polynomials is a critical skill in algebra that allows us to break down complex expressions into simpler, more manageable pieces. Essentially, factoring transforms a polynomial into a product of its factors, much like breaking down a number into its prime components. For example, a quadratic polynomial like \(ax^2 + bx + c\) can often be factored into the form \((px + q)(rx + s)\), where \(p\), \(q\), \(r\), and \(s\) are real numbers that satisfy the original quadratic when multiplied out.

To factor a polynomial, you look for common factors, use special product formulas (such as the difference of squares or perfect square trinomials), or apply techniques like grouping. In our exercise, factoring was the crucial step that allowed us to identify the common \((x-5)\) term, which was pivotal in simplifying the expression and moving closer to evaluating the limit.

Indeterminate Forms

In calculus, certain expressions are considered indeterminate forms because their limits can't be determined solely based on the operation involved. Examples of indeterminate forms include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \times \infty\), \(\infty - \infty\), \(0^0\), \(\infty^0\), and \(1^\infty\). These forms arise when applying limit theorems naively results in ambiguous or nonsensical mathematical statements.

Indeterminate forms require further manipulation to resolve what the actual limit is. Techniques such as factoring, conjugate multiplication, or L'Hôpital's Rule are often employed to transform these forms into ones that can be analyzed. In the exercise we examined, the form \(\frac{-(-2)}{0}\) represents an indeterminate type because the denominator approaches zero, which implies division by zero — a mathematical impossibility. This particular situation, where the function shoots off towards \(\pm\infty\) depending on the direction of approach, illustrates the need for careful examination of limits involving indeterminate forms.