Question

Determine the following limits. $$\lim _{w \rightarrow \infty} \frac{15 w^{2}+3 w+1}{\sqrt{9 w^{4}+w^{3}}}$$

Short answer

Answer: The limit of the given function as w approaches infinity is 5.

Step-by-step solution

Factor out the highest power of w from the numerator and denominator

First, factor out the highest power of w from both the numerator and the denominator. In this case, it's \(w^2\) in the numerator and \(w^2\) in the denominator. This will give: $$\lim _{w \rightarrow \infty} \frac{w^2(15 +\frac{3}{w} +\frac{1}{w^2})}{w^2\sqrt{9 +\frac{1}{w}}}$$

Simplify the expression

Cancel out \(w^2\) from the numerator and denominator. $$\lim _{w \rightarrow \infty} \frac{15 +\frac{3}{w} +\frac{1}{w^2}}{\sqrt{9 +\frac{1}{w}}}$$

Analyze the limit as w approaches infinity

As w approaches infinity, the terms \(\frac{3}{w}\) and \(\frac{1}{w^2}\) inside the numerator will approach 0, and the term \(\frac{1}{w}\) inside the denominator will also approach 0. Thus, we can rewrite the limit as: $$\lim _{w \rightarrow \infty} \frac{15}{\sqrt{9}}$$

Simplify and find the limit

Now, we can simply calculate the limit of the given expression. $$\lim _{w \rightarrow \infty} \frac{15}{\sqrt{9}} = \frac{15}{3} = 5$$ So, the limit of the given function as w approaches infinity is 5.

Key concepts

Infinite Limits

Understanding infinite limits is essential when evaluating the behavior of functions as they approach very large or very small values. In the context of the provided exercise, the concept of an infinite limit comes into play as we explore what happens to a function as the variable, in this case, w, heads towards infinity. This can tell us a lot about the end behavior of a function.

As w becomes very large, certain terms in the function will have less impact on the overall value. In the example, the limit is calculated as w approaches infinity. Terms like \( \frac{3}{w} \) and \( \frac{1}{w^2} \) approach zero as w increases, because the denominator grows much faster than the numerator. By focusing on the terms that dominate at large values of w, we find the infinite limits of our function, which, in this case, simplifies to a constant.

Limit Laws

Limit laws are the backbone of computing limits in a systematic and reliable way. They give us the ability to break down complex expressions into simpler parts that we can evaluate more easily. When applying limit laws, it is important to recognize when two expressions can be combined or split, and under which circumstances limits can be distributed across arithmetic operations.

In our exercise, the limit is simplified by separating the terms in the numerator and realizing that as w approaches infinity, some terms become negligible. This is an application of the limit laws, particularly the law that states that the limit of a sum is the sum of the limits (provided that the limits exist). Utilizing these laws, the step-by-step solution simplifies the limit problem and allows us to find that the function approaches a limit of 5 as w grows without bounds.

Rational Functions

A rational function is one that can be expressed as the ratio of two polynomials, and it's a common subject for limit problems in calculus. The exercise provided is an example of a rational function where the numerator is a polynomial of degree two and the denominator is a square root of a polynomial of degree four.

Rational functions often display interesting behaviors as the variable approaches infinity or specific values that may make the denominator zero. In the provided solution, we see the simplification process which is a vital technique in analyzing rational functions. After reducing the expression, we're left with a much simpler function that no longer depends on w, allowing us to easily find the limit as w approaches infinity. The appropriate handling of rational functions in limits can reveal the asymptotic behavior and other important characteristics of the function's graph.

Asymptotic Behavior

The asymptotic behavior of a function refers to how it acts as the input either grows very large or very small. In calculus, this often deals with determining what value a function approaches as the variable heads towards infinity or negative infinity, which tells us about the function's end behavior and can help us sketch its graph.

In the problem at hand, after simplifying the rational function, we determine that as w gets larger and larger, the function itself approaches 5. This means that the function has a horizontal asymptote at the line y = 5. Asymptotes give us a 'big picture' view of the function's behavior and are critical to understanding the long-term trends in the function's graph.