Problem 41 - Calculus Early Transcendentals

Question

Evaluate the following limits, where a and \(b\) are fixed real numbers. \(\lim _{x \rightarrow 4} \frac{x^{2}-16}{4-x}\)

Step-by-Step Solution

Answer

Answer: The limit of the function as x approaches 4 is -8.

Step 1: Write down the given function

Start by writing down the given function: \(\lim _{x \rightarrow 4} \frac{x^{2}-16}{4-x}\)

Step 2: Factor the numerator

Notice that the numerator is a difference of squares, which can be factored as follows: \(x^2 - 16 = (x + 4)(x - 4)\) Now, rewrite the function with the factored numerator: \(\lim _{x \rightarrow 4} \frac{(x + 4)(x - 4)}{4-x}\)

Step 3: Cancel out common factors

Cancel out the common factor in the numerator and the denominator, which is \((x - 4)\) and \(-(x - 4)\) Resulting in: \(\lim _{x \rightarrow 4} \frac{x + 4}{-1}\)

Step 4: Evaluate the limit

Now that the function is simplified, we can plug in x = 4 to evaluate the limit: \(\lim _{x \rightarrow 4} \frac{x + 4}{-1} = \frac{4 + 4}{-1} = \frac{8}{-1} = -8\) The limit of the given function as x approaches 4 is -8.

Key Concepts

Limits in Calculus

In the world of calculus, the concept of a limit is essential. It's the value that a function 'approaches' as the input (or 'x value') approaches a certain point. When evaluating limits, such as with the problem \(\lim _{x \rightarrow 4} \frac{x^2-16}{4-x}\), what we're really asking is 'What value does the expression tend towards as x gets infinitesimally close to 4?'

It's important to note that limits can exist even if the function itself is not defined at that specific point. This situation often occurs in cases where substituting the point directly into the function would result in an indeterminate form, like 0/0. Such forms tell us that we need to do a bit more work, like simplifying the expression, to find the limit's value. Understanding the concept of limits helps in analyzing the behavior of functions and is foundational to the study of calculus.

Difference of Squares

Factoring is a critical skill in algebra and calculus, especially when dealing with limits. One common type of factoring involves a 'difference of squares.' This is an expression like \(a^2 - b^2\), which can be factored into \(a+b\) and \(a-b\).

In our example, the numerator \(x^2 - 16\) is a difference of squares because it can be written as \(x^2 - 4^2\). We use the factoring formula to rewrite it as \(x+4)(x-4)\). Recognizing this pattern is key to simplifying expressions and finding limits, as it allows us to cancel out terms and avoid undefined expressions.

Factoring Polynomials

When you factor polynomials, you're breaking them down into products of simpler expressions. Factoring is not just a tool for solving equations but also for simplifying complex expressions in calculus. When you encounter a polynomial in the numerator or denominator while evaluating a limit, it's often necessary to factor it in order to simplify the expression to a form where the limit can be directly evaluated.

For example, the factoring step allowed us to transform \(x^2-16\) into \(x+4)(x-4)\), revealing a common factor with the denominator. This is a vital step because it can eliminate zero denominators or indeterminate forms, making it possible to compute the limit.

Simplifying Expressions

Simplifying expressions is the act of reducing a mathematical expression to its simplest form. This often involves factoring polynomials, canceling out terms, and combining like terms. In calculus, simplifying an expression is frequently an indispensable step when evaluating limits, as it helps to get rid of indeterminate forms and identify the behavior of a function as x approaches a particular value.

In our exercise, simplifying involved canceling the common \(x-4\) term in the numerator and denominator. This step was crucial: without it, we would be stuck with an undefined expression when substituting x=4. After simplifying, the expression became something we could easily evaluate by substituting x directly, leading us to the solution: the limit is -8.