Problem 67 - University Calculus: Early Transcendentals

Question

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow \infty} \frac{\sqrt{9 x+1}}{\sqrt{x+1}}$$

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Write the Limit Expression

We start by writing the given limit expression: \[\lim _{x \rightarrow \infty} \frac{\sqrt{9x+1}}{\sqrt{x+1}}\]Our goal is to find the limit as \(x\) approaches infinity.

Step 2: Simplify the Expression

We'll simplify the expression inside the limit by factoring out \(\sqrt{x}\) from both the numerator and the denominator.The expression becomes:\[\lim _{x \rightarrow \infty} \frac{\sqrt{x}\sqrt{9 + \frac{1}{x}}}{\sqrt{x}\sqrt{1 + \frac{1}{x}}}\] Notice how \(\sqrt{x}\) can be canceled out from the numerator and the denominator.

Step 3: Cancel Common Terms

Now, cancelling \(\sqrt{x}\) from the numerator and the denominator, we get:\[\lim _{x \rightarrow \infty} \frac{\sqrt{9 + \frac{1}{x}}}{\sqrt{1 + \frac{1}{x}}}\] This step reduces the expression significantly, clarifying the behavior of the expression as \(x\) approaches infinity.

Step 4: Evaluate the Limit

Note that as \(x\) goes to infinity, \(\frac{1}{x}\) approaches zero. Therefore, the expression simplifies to:\[\lim _{x \rightarrow \infty} \frac{\sqrt{9 + 0}}{\sqrt{1 + 0}} = \frac{\sqrt{9}}{\sqrt{1}} = \frac{3}{1} = 3\]Hence, the limit is 3.

Key Concepts

Simplifying expressions

When dealing with complex mathematical expressions, simplifying them can make finding the solution much clearer. In limits, especially when involving square roots or other intricate functions, simplifying expressions helps reveal underlying patterns or behaviors. For example, in the step-by-step solution provided, simplifying helps us tackle the limit as \(x\) approaches infinity by factoring out common terms. This is a starting point to understanding how the function behaves.

• Identify similar terms: Recognizing components of the expression that repeat can allow you to factor and simplify.
• Factor common elements: Taking out common factors, like \(\sqrt{x}\) in the example, often makes these terms easier to cancel and interpret.
• Cancel and reduce: By cancelling common terms, you streamline the expression, making it much easier to evaluate as \(x\) changes.

Breaking down problems in this way reduces complexity and facilitates a clearer path toward finding a solution.

Infinity in calculus

Infinity in calculus is a fascinating concept since it denotes a number larger than any other conceivable number. It's crucial to understand how functions behave as we approach infinity to find limits effectively. In the context of the exercise, we explore what happens as \(x\) grows indefinitely.

• Approaching infinity: In many limit problems, as \(x\) approaches infinity, specific terms either disappear (like \(\frac{1}{x}\) approaching zeros) or dominate.
• Growth rates: Identifying how terms in a function grow relative to one another helps determine how they transform as \(x\) increases.
• Simplification impact: As variables like \(x\) head towards these extreme values, simplified expressions showcase actual impacts of each term better.

This understanding is very significant when simplifying and solving expressions where infinity plays a central role, such as in this problem.

Factoring in calculus

Factoring, often used in algebra, also plays a pivotal role in calculus. When tackling limits, factoring can simplify the expression and expose relationships between terms. In calculus, it's not only about dealing with polynomial equations but also simplifying under roots and fractions.

• Identify common factors: Spotting terms that can be factored out or extracted aids in reducing the complexity of a function.
• Reduce to essentials: By factoring out \(\sqrt{x}\), as done in the solution, the expression becomes more manageable and intuitive.
• Enhance understanding: This process often leads to discovering how the separate parts of a function behave together as a whole, especially near extremes like infinity.

Factoring is thus a bridge from a complex-looking expression to its simpler, core components, easing the process of evaluating limits and other calculus operations.