Problem 7 - 5 Steps to A 5: AP Calculus AB 2021

Question

Find the limits of the following: \(\lim _{x \rightarrow-\infty} \frac{3 x}{\sqrt{x^{2}-4}}\)

Step-by-Step Solution

Answer

Answer: The limit is 3.

Step 1: Simplify the Function

Firstly, we simplify the given function: \(\frac{3x}{\sqrt{x^2 - 4}}\)

Step 2: Identify Dominant Terms

As x approaches negative infinity, the dominant terms are the ones with the highest power of x. In this case, the dominant term in the numerator is \(3x\), and the dominant term in the denominator is \(\sqrt{x^2}\).

Step 3: Divide by Highest Power of x in Denominator

To find the limit, we need to get a standard form of the given function. To do this, we divide both the numerator and denominator by the highest power of x present in the denominator: \(\frac{(3x)/(x)}{(\sqrt{x^2 - 4})/(x)} = \frac{3}{\sqrt{\frac{x^2 - 4}{x^2}}}\)

Step 4: Simplify and Find the Limit

Now, let's simplify the function further and find its limit as x approaches negative infinity: For the denominator, we know that: \(\lim_{x \rightarrow -\infty} \frac{x^2 - 4}{x^2} = \lim_{x \rightarrow -\infty} \frac{x^2}{x^2} - \frac{4}{x^2} = 1 - 0 = 1\) So the simplified function becomes: \(\frac{3}{\sqrt{1}} = 3\) As x approaches negative infinity, the limit of the function is 3. So, we have: \(\lim_{x \rightarrow -\infty} \frac{3x}{\sqrt{x^2 - 4}} = 3\)

Key Concepts

Limit at Infinity

When dealing with calculus, especially in AP Calculus AB, understanding limits at infinity is crucial. A limit at infinity asks what happens to a function as its input goes to a positive or negative infinity. This is different from approaching a specific value, and it involves the "end behavior" of functions.

For the given exercise, \(\lim_{x \rightarrow -\infty} \frac{3x}{\sqrt{x^{2}-4}}\), we want to determine what happens as \(x\) becomes infinitely large in the negative direction. This will tell us the behavior of the function at that extreme. In practical terms, limits at infinity help us understand how a function behaves in the very long-run, beyond any finite range. By simplifying each term in the function through observing its growth rate, we can find the limit and thus its horizontal asymptote, if it exists.

Rational Functions

Rational functions are expressions that can be modeled as the ratio of two polynomials. In other words, any function where an equation is given in the form \(\frac{P(x)}{Q(x)}\).

In our exercise, the function \(\frac{3x}{\sqrt{x^{2}-4}}\) isn't a standard rational function because the denominator has a square root. Yet, it still behaves similarly, especially as we explore limits and asymptotic behavior.

• Numerator: A polynomial function, in this case, \(3x\).
• Denominator: Involves a square root of a polynomial, \(\sqrt{x^2 - 4}\).

When evaluating such functions at infinity, the presence of radicals doesn't fundamentally change the approach, but it does require careful handling to simplify the expression and evaluate the limit.

Dominant Terms

In calculus, especially when evaluating limits at infinity, dominant terms play a central role. Dominant terms are those parts of the function that grow faster than others, determining the overall behavior of the function.

In the exercise, the dominant term in the numerator is \(3x\), while in the denominator, it is \(\sqrt{x^2}\). Both these terms have the most substantial impact as \(x\) goes to negative infinity since they contain the highest powers of \(x\). By focusing on these, you can simplify complex expressions into simpler ones that are easier to evaluate. This strategy allows us to reduce the original problem to just finding the limit of the dominant terms, thus simplifying the calculation and making the solution more straightforward.

Asymptotic Behavior

Understanding asymptotic behavior is a crucial aspect of mathematics, particularly in calculus. It describes how functions behave as inputs either get very large or very small.

For our rational function, \(\lim_{x \rightarrow -\infty} \frac{3x}{\sqrt{x^2 - 4}} = 3\), we see that as \(x\) approaches negative infinity, the function approaches the line \(y = 3\) horizontally.

• Horizontal Asymptotes: Lines that the graph of the function approaches as \(x\) becomes large (positively or negatively).

Asymptotic behavior gives us a simple way to predict and describe the ultimate behavior of a function without needing to calculate every specific value. This knowledge is incredibly helpful for understanding complex functions in a more intuitive way. Overall, recognizing asymptotic behavior leads to valuable insights into how a given function might behave across its entire domain.