Problem 53 - Precalculus

Question

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

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{4}\).

Step 1: Plug in the Limit

First, substitute the value of \( x \) approaching 1 into the expression: \[ \frac{\frac{1}{1+1}-1^{2}}{\frac{1}{1}+2 \times 1^{2}-1} = \frac{\frac{1}{2} - 1}{1 + 2 - 1} \]. This results in \( \frac{-\frac{1}{2}}{2} \).

Step 2: Simplify the Expression

Simplify the resulting fraction from Step 1: \[ \frac{-\frac{1}{2}}{2} = -\frac{1}{4} \].

Step 3: Conclude the Limit

After simplification, we conclude that the limit of the expression as \( x \) approaches 1 is \( -\frac{1}{4} \).

Key Concepts

Limit Properties

The study of limits is fundamental in calculus and offers vital tools for analyzing how functions behave as inputs approach particular points or even infinity. Limit properties are rules that make it easier to calculate limits without extensive computation. These properties allow us to manage complex functions by breaking them down into simpler parts. Some basic limit properties include:

• Sum/Difference property: The limit of the sum/difference of two functions is the sum/difference of their limits.
• Product property: The limit of a product of two functions is the product of their limits.
• Quotient property: The limit of a quotient of two functions is the quotient of their limits, given that the limit of the denominator is not zero.

In the exercise above, employing these properties helps simplify the calculation by allowing substitution and re-organization of terms. This saves us from directly grappling with complex expressions.

Substitution Method

The substitution method is a common way to find limits, especially when the function is already simplified enough. By substituting a value directly into the function, we can often quickly find the limit. In our specific exercise, the method involves simply putting the approaching value of \( x \) into the expression:

• Calculate the individual components separately. Here, substitute \( x = 1 \) into the expression within the numerator and denominator separately.
• The substitutions render the complex fraction: \( \frac{\frac{1}{2} - 1}{1 + 2 - 1} \).
• Evaluate these simpler expressions separately for cleaner results.

This approach is particularly powerful when the function does not produce an indeterminate form like \( \frac{0}{0} \), allowing for straightforward calculation.

Simplifying Expressions

Simplification is often necessary to make the calculation of limits easy. In essence, it revolves around reducing a complex expression into a simpler form. This can be done in various ways, depending on what the expression contains.In the exercise given, after substitution, we are left with the expression \( \frac{-\frac{1}{2}}{2} \). Here are the steps involved in simplification:

• First, simplify the numerator: \(-\frac{1}{2}\).
• Then, simplify the entire fraction by dividing the numerator by the denominator: \( -\frac{1}{4} \).

This kind of simplification not only makes the result easier to interpret but also reveals the behavior of the function as it approaches the limit value. The intention is to reduce any chances of error and make the final result more apparent and trustworthy.