Question

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 4} \frac{x-4}{x^{2}-16} $$

Short answer

The limit of the function as x approaches 4 is \( \frac{1}{8} \).

Step-by-step solution

Simplify the Function

First, try to simplify the function as much as possible. You can begin by factoring the denominator, giving you the following: \( \frac{x-4}{(x-4)(x+4)} \). This simplification allows us to cancel out the repeating term \(x-4\) from the numerator and denominator.

apply the limit

After canceling out, we obtain: \( \frac{1}{x+4} \). Now we can directly substitute the given limit value, which is 4 in this exercise, into the simplified function. Remember the rule that if you obtain a finite value after substitution, then that value is the limit.

compute the limit

After substituting 4 into the function we get: \( \frac{1}{4+4} = \frac{1}{8} \).

Key concepts

Understanding the Limit of a Function

When delving into calculus, 'limit of a function' is a fundamental concept you'll encounter. In essence, a limit tries to describe the behavior of a function as the input approaches a certain value, but without necessarily reaching that value. This insight is vital because it helps understand functions at points where they might not be well-defined, like in cases of division by zero.

For instance, in the exercise \(\lim _{x \rightarrow 4} \frac{x-4}{x^{2}-16}\), we're interested in finding what happens to the function as \(x\) gets closer and closer to 4. A limit does not always exist, but when it does, as in this scenario, it tells us the value that \(y\) (the function's output) is approaching. The step-by-step solution illustrates how the function simplifies and how the actual value is determined, paving the way to understanding continuity and predicting function behavior across its domain.

The Role of Factoring Polynomials in Finding Limits

Factoring polynomials is a crucial algebraic skill that comes in handy when computing limits in calculus. It involves breaking down polynomials into products of their factors, which can simplify solving equations or, as for the limits, finding the behavior of a function around specific points.

In the given example, the denominator \(x^2-16\) is factored into \((x-4)(x+4)\), which reveals the repeating \(x-4\) term that also appears in the numerator. This common factor can be cancelled out, thus transforming the original complex fraction into a much simpler form. Without factoring, identifying such simplifications would be considerably more challenging, potentially complicating the limit finding process. Hence, mastering factoring polynomials is a must for anyone dealing with limits.

Applying Limit Substitution Method

The 'limit substitution' strategy is a shortcut that can be employed when the function is continuous at the point to which \(x\) is approaching—in other words, if directly substituting the value of \(x\) yields a finite number. After simplifying the function, like in the provided example \(\frac{1}{x+4}\), we can then substitute \(x\) with the limit we're approaching—here, 4.

Implementing the Substitution

Provided the expression is simplified, you will get \(\frac{1}{4+4} = \frac{1}{8}\). Substitution provides a quick and often intuitive way to find limits for functions that are well-behaved without the need for more complex calculus techniques.

However, bear in mind this method only works when there are no discontinuities (such as holes or vertical asymptotes) at the point of interest, ensuring the smooth behavior of the function.