Problem 10 - Calculus with Concepts in Calculus

Question

Find the given limit. $$ \lim _{x \rightarrow-\infty} \frac{x-\frac{1}{2}}{\frac{1}{2} x+1} $$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Identify the Limit Type

We are asked to find the limit of the function \( f(x) = \frac{x - \frac{1}{2}}{\frac{1}{2}x + 1} \) as \( x \) approaches \( -\infty \). This is a rational function where the highest power terms in numerator and denominator dominate the behavior as \( x \to -\infty \).

Step 2: Simplify the Function

To solve the limit, we can simplify the expression by dividing each term in the numerator and the denominator by \( x \), the highest power term.So,\[ \frac{x - \frac{1}{2}}{\frac{1}{2}x + 1} = \frac{x\left(1 - \frac{1}{2x}\right)}{x\left(\frac{1}{2} + \frac{1}{x}\right)} \]This simplifies to,\[ \frac{1 - \frac{1}{2x}}{\frac{1}{2} + \frac{1}{x}} \].

Step 3: Evaluate the Limit

As \( x \to -\infty \), the terms \( \frac{1}{2x} \) and \( \frac{1}{x} \) both approach 0. Therefore, the expression simplifies to:\[ \frac{1 - 0}{\frac{1}{2} + 0} = \frac{1}{\frac{1}{2}} \].

Step 4: Calculate the Result

The simplified expression \( \frac{1}{\frac{1}{2}} \) further simplifies by multiplying numerator and denominator by 2 to get 2. Thus, the limit of the function as \( x \to -\infty \) is 2.

Key Concepts

Rational Functions

Rational functions are mathematical expressions defined as the ratio of two polynomials. These functions can be written in the form \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).

• The degree of the polynomial is the highest power of \( x \) present in the expression.
• Rational functions often have points of discontinuity where the denominator equals zero.
• Understanding the degree of both the numerator and the denominator helps determine the behavior of the function as \( x \to \pm \infty \).

In the given exercise, we have \( f(x) = \frac{x - \frac{1}{2}}{\frac{1}{2} x + 1} \). Both the numerator and denominator are polynomials where the highest degree is 1, making this a simple, linear rational function. This means, as \( x \) approaches infinity in either direction, the asymptotic behavior relates primarily to these highest power terms.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the input, in this case \( x \), moves towards a certain limit such as infinity. For rational functions, the highest power term in both the numerator and the denominator often determines this behavior:

• If the degrees of the numerator and the denominator are equal, the horizontal asymptote is found by dividing the leading coefficients.
• If the degree of the numerator is less than that of the denominator, the horizontal asymptote is the x-axis, \( y = 0 \).
• If the degree of the numerator is greater, the rational function will not have a horizontal asymptote but will have an oblique (slant) asymptote.

In our exercise, since the degrees of both the numerator \( x \) and the denominator \( \frac{1}{2}x \) are the same, the asymptotic behavior is determined by the coefficients. Here, as \( x \to -\infty \), the function behaves similarly to \( \frac{x}{\frac{1}{2}x} = 2 \), implying a horizontal asymptote at \( y = 2 \).

Simplifying Expressions

Simplifying expressions is a critical step to make evaluation of limits more manageable. Especially with rational functions, simplification will often reveal the core behavior of the function:

• Divide every term in both the numerator and the denominator by the highest power of \( x \).
• This method reduces the expression by canceling terms that approach zero as \( x \to \pm \infty \).
• Resultant expressions typically involve fewer terms, making the end behavior more obvious.

In our problem, we simplify \( \frac{x - \frac{1}{2}}{\frac{1}{2}x + 1} \) by dividing each component by \( x \). This leads to a reduced form \( \frac{1 - \frac{1}{2x}}{\frac{1}{2} + \frac{1}{x}} \). As \( x \to -\infty \), terms like \( \frac{1}{2x} \) and \( \frac{1}{x} \) decay to zero, simplifying our function to \( \frac{1}{\frac{1}{2}} = 2 \). This approach not only makes calculating the limit simpler but also gives a clear view of the function's long-term behavior.