Problem 63 - Precalculus

Question

Evaluate the limits using the limit properties. $$\lim _{x \rightarrow 3} \frac{\frac{2}{x+1}-3 x}{\sqrt{x-2}+1}$$

Step-by-Step Solution

Answer

The limit is \( \frac{-17}{4} \).

Step 1: Substitute the Limit Value

First, attempt to substitute the limit value, which is 3, directly into the expression: \( \lim _{x \rightarrow 3} \frac{\frac{2}{x+1}-3 x}{\sqrt{x-2}+1} \). This becomes \( \frac{\frac{2}{3+1}-3 \times 3}{\sqrt{3-2}+1}\).

Step 2: Simplify the Expression

Simplify each part of the expression: \( \frac{2}{3+1} = \frac{2}{4} = \frac{1}{2} \), \( 3 \times 3 = 9 \), and \( \sqrt{3-2} = \sqrt{1} = 1 \). The expression becomes \( \frac{\frac{1}{2} - 9}{1 + 1} = \frac{\frac{1}{2} - 9}{2}\).

Step 3: Further Simplification

Combine \( \frac{1}{2} - 9 \) by converting 9 into \( \frac{18}{2} \). Now subtract: \( \frac{1}{2} - \frac{18}{2} = \frac{-17}{2} \). The expression becomes \( \frac{\frac{-17}{2}}{2} = \frac{-17}{4} \).

Step 4: Evaluate the Limit

Since we have successfully substituted the values and simplified the expression, the evaluated limit is \( \frac{-17}{4} \).

Key Concepts

Limit Properties

In calculus, understanding the properties of limits is crucial for evaluating functions as they approach certain points. Various rules help simplify complex expressions:

• Sum Rule: The limit of a sum is the sum of the limits.
• Difference Rule: The limit of a difference is the difference of the limits.
• Product Rule: The limit of a product is the product of the limits.
• Quotient Rule: The limit of a quotient is the quotient of the limits, provided the denominator is not zero.

Each of these properties can be applied step by step in complex expressions to make calculating limits more manageable. By breaking down expressions into smaller, simpler parts, you can evaluate limits more easily as shown in our exercise.

Direct Substitution in Limits

Direct substitution is often the first step when evaluating a limit. The idea here is straightforward: substitute the value that the variable is approaching directly into the expression.
If this gives a determinate number, the limit is found. For example, in our exercise, plugging in 3 initially looks promising. But sometimes, direct substitution results in an undefined form like (\(\frac{0}{0}\)), indicating further steps are needed.
It's essential once you've substituted to simplify and check if the expression itself can actually be evaluated through direct substitution. This can quickly confirm whether a limit can be directly solved or requires simplification.

Simplification of Expressions

Simplifying expressions is a fundamental skill in calculus, especially with limits. Once direct substitution is applied, it often helps to simplify the expression for clearer insight.
In our example, simplifying the fractions and radicals involved, like (\(\frac{2}{3+1}\)) to (\(\frac{1}{2}\)), and dealing with (\(3 \times 3\)) or (\(\sqrt{3-2}\)) to 1, make the process straightforward.

• Simplified Fractions: Dealing with easy numbers reduces complexity.
• Radicals Simplification: Converting to whole numbers whenever possible aids understanding.

This process highlights any indeterminate forms that require further manipulation. Simplification makes it easier to apply limit rules and recognize the necessary operations to evaluate the expression.

Evaluating Limits

After simplifying, the final step is to evaluate the limit. This process involves thoroughly simplifying all parts leading up to writing down the evaluated limit.
In the exercise, having broken down each component, the limit (\(\lim _{x \rightarrow 3}\)) eventually simplifies to (\(\frac{-17}{4}\)).
Evaluating is about confirming whether the expression approaches a specific value as the variable gets infinitely close to the given number. This understanding is the keystone of calculus differentiation and integration processes, manifesting in how we tackle real-world changes and equations.
Ultimately, evaluating limits is about analyzing control points and applying limit rules, leading you to the correct solution.