Problem 35 - Contemporary Precalculus

Question

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty} \frac{x^{2}+2 x+1}{\sqrt{x^{4}+2 x}}$$

Step-by-Step Solution

Answer

Short Answer: When applying the Infinite Limit Theorem and simplifying the given function, the limit as x approaches infinity of the function \(\frac{x^{2}+2x+1}{\sqrt{x^{4}+2x}}\) is equal to 0.

Step 1: Identify Highest Power of x

We start by identifying the highest power of x present in the given function, which in this case is x^4 in the denominator. We will use this to simplify the function.

Step 2: Divide Both Numerator and Denominator by Highest Power of x

Next, we need to divide both the numerator and the denominator of the given function by the highest power of x, in this case \({(x^4)}\). Doing so, we get: $$\lim _{x \rightarrow \infty} \frac{\frac{x^{2}}{x^{4}}+\frac{2x}{x^{4}}+\frac{1}{x^{4}}}{\sqrt{\frac{x^{4}}{x^{4}}+\frac{2x}{x^{4}}}}$$ This simplifies to: $$\lim _{x \rightarrow \infty} \frac{\frac{1}{x^{2}}+\frac{2}{x^{3}}+\frac{1}{x^{4}}}{\sqrt{1+\frac{2}{x^{3}}}}$$

Step 3: Apply Infinite Limit Theorem

Now, as x approaches infinity, any term with x in the denominator tends to zero. Applying the Infinite Limit Theorem, we can simplify the given limit: $$\lim _{x \rightarrow \infty} \frac{0+0+0}{\sqrt{1+0}}$$

Step 4: Compute the Limit

The simplified expression is: $$\lim _{x \rightarrow \infty} \frac{0}{\sqrt{1}}$$ Since the numerator is 0 and the denominator is a non-zero constant value, the limit is equal to 0: $$\lim _{x \rightarrow \infty} \frac{x^{2}+2 x+1}{\sqrt{x^{4}+2 x}} = 0$$

Key Concepts

Infinite Limit Theorem

The Infinite Limit Theorem is a foundational concept in calculus that deals with the behavior of functions as they approach infinity. When the independent variable within a function tends towards infinity, the function itself may approach a specific value, which we call the limit at infinity. To determine this limit, if it exists, we often simplify the function by dividing through by the highest power of the variable in the expression.

For instance, to analyze the limit of the function \( \lim _{x \rightarrow \infty} \frac{x^{2}+2 x+1}{\sqrt{x^{4}+2 x}} \), we observe the highest power of \(x\) in the denominator and divide both the numerator and the denominator by \(x^4\). This method effectively reveals the dominant terms that determine the function's behavior as \(x\) grows larger. Terms involving powers of \(x\) in the denominator tend to zero as per the theorem, which simplifies our expression considerably when \(x\) approaches infinity.

Properties of Limits

Limits in calculus follow specific properties that enable us to perform algebraic manipulations and reach a solution. One such rule is that the limit of a sum equals the sum of the limits, provided the individual limits exist. Similarly, the limit of a product is the product of the limits. In our case, we can separately evaluate the limit of each term in the function. When faced with a rational expression, one useful approach is to divide through by the highest power of \(x\) present in the denominator before proceeding. Doing so allows each term to be simplified, and those involving \(x\) in the denominator will trend towards zero, as \(x\) approaches infinity. This is particularly crucial in problems like \( \lim _{x \rightarrow \infty} \frac{x^{2}+2 x+1}{\sqrt{x^{4}+2 x}} \), where after applying this property, each term in the numerator containing \(x\) becomes insignificant compared to the constant terms as \(x\) increases without bound.

Asymptotic Behavior

When studying limits in calculus, we often encounter functions that have asymptotic behavior. This refers to the tendency of a function to get closer and closer to a particular line, called an asymptote, as the input either increases or decreases without bound.

An asymptote may be horizontal, vertical, or even oblique. For the function \( \frac{x^{2}+2 x+1}{\sqrt{x^{4}+2 x}} \) as \(x\) approaches infinity, we're looking at a horizontal asymptote, which is a constant value that the function will approach but never quite reach. In the given problem, as \(x\) becomes larger, the terms that decrease in magnitude due to division by a power of \(x\) become less significant, and the function's value gets closer to zero. Thus, the horizontal asymptote would be the line \(y=0\), illustrating a key concept in understanding asymptotic behavior in the context of limits.