Problem 64 - Precalculus

Question

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

Step-by-Step Solution

Answer

The limit is \( \frac{4}{3} \).

Step 1: Understand the Problem

Evaluate the given limit \( \lim _{x \rightarrow-1} \frac{\left(\frac{x}{2 x+1}\right)^{2}-3 x}{\sqrt{x^{2}+3}+1} \). This involves substituting \( x = -1 \) into the function and identifying any indeterminate forms to apply limit properties or algebraic manipulation.

Step 2: Substitute Directly

Substitute \( x = -1 \) directly into the function: \[ f(x) = \frac{\left(\frac{-1}{2(-1)+1}\right)^{2} - 3(-1)}{\sqrt{(-1)^{2}+3} + 1}. \]Simplifying, we get:\[ f(-1) = \frac{\left(\frac{-1}{-1}\right)^{2} + 3}{\sqrt{1+3} + 1} = \frac{1 + 3}{2 + 1} = \frac{4}{3}. \] Since direct substitution does not yield an indeterminate form, the limit can be found easily.

Step 3: Conclusion

Since substituting \( x = -1 \) directly into the equation doesn't yield an indeterminate form, the limit is simply the function evaluated at \( x = -1 \). Thus, the limit is \( \frac{4}{3}. \)

Key Concepts

Limit Properties

Understanding limit properties is crucial when evaluating limits. These properties help simplify complex expressions and manage indeterminate forms. Here are a few key properties:

• Sum/Difference Rule: The limit of a sum/difference is the sum/difference of the limits.
• Product/Quotient Rule: The limit of a product/quotient is the product/quotient of the limits, provided the limit of the denominator isn't zero.
• Power/Root Rule: The limit of a power/root is the power/root of the limit.

In the exercise, these properties allow us to rearrange and simplify the given function. By recognizing these properties, calculations become straightforward, even if first appearances suggest complexity.

Substitution Method

The substitution method is a straightforward approach to evaluating limits when no indeterminate form, such as 0/0, arises. This method directly substitutes the target value into the function. If this substitution doesn't cause any undefined terms, it often quickly reveals the limit.

In the exercise example, substituting directly with \(x = -1\) doesn't produce an indeterminate form. This means the substitution method effectively simplifies the calculation to mere evaluation of expression, allowing us to find the solution without requiring more complex algebraic manipulation or additional properties. By checking for directly applicable substitutes, you can often save time and effort when evaluating limits.

Function Evaluation

Function evaluation is the process of substituting a specific value into a function to obtain a result. It ties closely with the substitution method in limit problems. Here, it helps validate that the result obtained doesn't involve any undefined operations.

In this example, replacing \(x = -1\) directly into the function leads to easy simplification. We calculate each part:

• Evaluate the numerator: \(\left(\frac{-1}{2(-1)+1}\right)^2 - 3(-1)\)
• Evaluate the denominator: \(\sqrt{(-1)^2 + 3} + 1\)

This showcases how direct function evaluation provides verification, ensuring the limit isn't caught in undefined behavior like division by zero. Consequently, it guides students to confirm their answers confidently.