Problem 38 - Contemporary Precalculus

Question

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty} \frac{\sqrt{x}+2}{\sqrt{x}-3}$$

Step-by-Step Solution

Answer

Answer: The limit of the given function as x goes to infinity is 1.

Step 1: Rewrite the function

First, rewrite the given function in a way that isolates the x terms to simplify limit computation. $$ \lim_{x\rightarrow\infty} \frac{\sqrt{x}+2}{\sqrt{x}-3} $$ To isolate the x terms, divide both numerator and denominator by \(\sqrt{x}\): $$ \lim_{x\rightarrow\infty} \frac{\frac{\sqrt{x}}{\sqrt{x}}+\frac{2}{\sqrt{x}}}{\frac{\sqrt{x}}{\sqrt{x}}-\frac{3}{\sqrt{x}}} $$

Step 2: Simplify the function

Simplify the expression by performing the calculations in the numerator and the denominator. $$ \lim_{x\rightarrow\infty} \frac{1+\frac{2}{\sqrt{x}}}{1-\frac{3}{\sqrt{x}}} $$

Step 3: Apply Limit Properties

Now we can use the properties of limits and Infinite Limit Theorem to find the limit. Take the limit of each component separately: $$ \lim_{x\rightarrow\infty} 1 = 1 $$ $$ \lim_{x\rightarrow\infty} \frac{2}{\sqrt{x}} = 0 $$ $$ \lim_{x\rightarrow\infty} \frac{3}{\sqrt{x}} = 0 $$

Step 4: Substituting the limit

Substitute the limits of each component back into the expression: $$ \lim_{x\rightarrow\infty} \frac{1+0}{1-0} $$

Step 5: Solve the limit

Calculate the final limit: $$ \lim_{x\rightarrow\infty} \frac{1}{1} = 1 $$ Therefore, the limit of the given function as \(x\) goes to infinity is equal to 1.

Key Concepts

Limits in Calculus

Understanding the fundamental concept of limits in calculus is like opening a door to the world of continuous mathematics. Limits help us determine the value that a function approaches as the input approaches some point. In other words, a limit describes the behavior of a function as we get closer and closer to a certain input value, without necessarily reaching it.

Take, for example, the expression \( \lim_{x \rightarrow a} f(x) \), which reads as 'the limit of the function \( f \) as \( x \) approaches \( a \).' When the value of \( x \) gets infinitely close to \( a \)—\( x \) could be approaching from either direction on the number line—what value does the function \( f(x) \) approach?

This is what the limit seeks to define, and we use it to understand notions such as continuity, derivatives, and integrals—cornerstones of calculus that have applications in physics, engineering, economics, and beyond. Specifically, limits allow us to calculate the instantaneous rate of change (derivative) of functions, and the total accumulation (integral) under a curve—concepts that are nearly impossible to grasp with simple algebra alone.

Properties of Limits

Diving deeper into calculus, we come across properties of limits which are rules that simplify the computation of limits. These properties are not arbitrary but are grounded in logical mathematics and give us tools for solving complex problems with relative ease.

Some of the most important properties include:

• The limit of a constant is the constant itself.
• The limit of a sum is the sum of the limits, provided the limits exist.
• The limit of a product is the product of the limits, under the same existence condition.
• The limit of a quotient is the quotient of the limits, assuming the limit of the denominator is not zero.
• Limits can be distributed across powers and roots, if those limits exist.

Each of these properties allows for step-by-step simplification of complex functions so that determining the limit becomes more manageable. As in our example, these properties help us to dismantle \( \lim_{x \rightarrow \infty} \frac{\sqrt{x}+2}{\sqrt{x}-3} \) into simpler parts to understand how the function behaves as \( x \) goes to infinity.

Solving Limits Step by Step

When confronted with a limit problem, a systematic approach is your best ally. Let's look at how the steps apply to our example, \( \lim_{x \rightarrow \infty} \frac{\sqrt{x}+2}{\sqrt{x}-3} \), illustrating how to solve for the limit step by step.

Rewriting the Function

First, we rewrite the function to isolate terms involving \( x \) to simplify our work. In our case, both the numerator and denominator have \( \sqrt{x} \), which we use to our advantage by dividing each term by \( \sqrt{x} \).

Simplifying the Function

Next, we simplify the resulting expression, performing algebra on the simplified terms. This reduction often reveals the behavior of the function as \( x \) grows large.

Applying Limit Properties

We apply limit properties, breaking down the function into parts for which we can more easily compute limits. Sometimes, this involves understanding the functions' behavior at infinity or at a specified point.

Substituting Component Limits

After determining individual limits of components, we substitute those back into the function. In many cases, this reveals indeterminate forms that further require techniques like l'Hôpital's Rule or rationalization.

Solving the Final Limit​

Finally, we complete the calculation for the overall limit. With our example, we land on the understanding that as \( x \) approaches infinity, our function converges to the value of 1. The step-by-step approach takes the mystique out of the limit and shows the clear path to the solution.