Horizontal Asymptote For F(x) = (2x^2+2)/(2x^2-2)

When dealing with rational functions, one of the key features we often look for is the presence of horizontal asymptotes. These lines represent the behavior of the function as the input values ($x$) approach positive or negative infinity. Essentially, they tell us where the graph of the function is heading in the long run. For the given function, f(x)=rac{2 x^2+2}{2 x^2-2}, we're going to dive deep into determining its horizontal asymptote, if one exists. This involves a systematic approach based on comparing the degrees of the polynomial in the numerator and the denominator. Understanding this concept is crucial for sketching accurate graphs of rational functions and for a comprehensive understanding of their end behavior.

Understanding Horizontal Asymptotes

A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as $x$ tends towards positive or negative infinity. It's like a guiding line that the function gets closer and closer to, but may or may not actually touch. For rational functions, which are ratios of two polynomials, the existence and location of horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator. Let's denote the numerator polynomial as $P(x)$ and the denominator polynomial as $Q(x)$, so our rational function is in the form f(x) = rac{P(x)}{Q(x)}. We need to consider three cases based on the degrees of these polynomials:

1. Degree of $P(x)$ is less than the degree of $Q(x)$: In this scenario, the horizontal asymptote is always the line $y=0$. This is because as $x$ gets very large, the denominator grows much faster than the numerator, making the entire fraction approach zero.

2. Degree of $P(x)$ is equal to the degree of $Q(x)$: Here, the horizontal asymptote is the line y = rac{a}{b}, where $a$ is the leading coefficient of the numerator polynomial and $b$ is the leading coefficient of the denominator polynomial. The function approaches the ratio of these leading coefficients as $x$ goes to infinity.

3. Degree of $P(x)$ is greater than the degree of $Q(x)$: In this case, there is no horizontal asymptote. Instead, the function may have a slant (or oblique) asymptote or grow without bound. The end behavior is not a horizontal line.

It's important to remember that a function can cross its horizontal asymptote. The asymptote describes the behavior as $x$ approaches infinity, not necessarily for finite values of $x$. However, for rational functions, once the degree of the denominator is greater than or equal to the degree of the numerator, the function will not cross the horizontal asymptote for large absolute values of $x$.

Analyzing the Given Function: f(x)=rac{2 x^2+2}{2 x^2-2}

Now, let's apply these rules to our specific function: f(x)=rac{2 x^2+2}{2 x^2-2}. First, we need to identify the numerator and the denominator polynomials and their respective degrees.

• The numerator is $P(x) = 2x^2 + 2$. The highest power of $x$ in the numerator is $x^2$, so the degree of the numerator is 2.
• The denominator is $Q(x) = 2x^2 - 2$. The highest power of $x$ in the denominator is also $x^2$, so the degree of the denominator is 2.

We are in the second case described above: the degree of the numerator is equal to the degree of the denominator. This is a key observation that guides us to the correct horizontal asymptote.

Determining the Horizontal Asymptote

Since the degrees of the numerator and the denominator are equal (both are 2), the horizontal asymptote is determined by the ratio of the leading coefficients of these polynomials.

• The leading coefficient of the numerator ($2x^2 + 2$) is 2.
• The leading coefficient of the denominator ($2x^2 - 2$) is 2.

To find the horizontal asymptote, we divide the leading coefficient of the numerator by the leading coefficient of the denominator:

y = rac{ ext{Leading coefficient of numerator}}{ ext{Leading coefficient of denominator}} = rac{2}{2}

$$y = 1$$

Therefore, the horizontal asymptote for the function f(x)=rac{2 x^2+2}{2 x^2-2} is the line $y=1$. This means that as $x$ approaches positive infinity ($x o ext{ } extbf{ } ext{ } extbf{ }$) or negative infinity ($x o - ext{ } extbf{ } extbf{ } extbf{ } $), the value of $f(x)$ will get closer and closer to 1.

Alternative Method: Limit Evaluation

Another rigorous way to find the horizontal asymptote is by evaluating the limit of the function as $x$ approaches infinity. This method directly applies the definition of a horizontal asymptote.

We want to find:

$$\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{2x^2+2}{2x^2-2}$$

To evaluate this limit, we can divide every term in the numerator and the denominator by the highest power of $x$ present in the denominator, which is $x^2$:

$$\lim_{x \to \infty} \frac{\frac{2x^2}{x^2} + \frac{2}{x^2}}{\frac{2x^2}{x^2} - \frac{2}{x^2}} = \lim_{x \to \infty} \frac{2 + \frac{2}{x^2}}{2 - \frac{2}{x^2}}$$

Now, as $x$ approaches infinity, the terms rac{2}{x^2} approach 0:

$$\frac{2 + 0}{2 - 0} = \frac{2}{2} = 1$$

So, $\lim_{x \to \infty} f(x) = 1$.

Similarly, we can evaluate the limit as $x$ approaches negative infinity:

$$\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} \frac{2x^2+2}{2x^2-2}$$

Dividing by $x^2$ again:

$$\lim_{x \to -\infty} \frac{2 + \frac{2}{x^2}}{2 - \frac{2}{x^2}} = \frac{2 + 0}{2 - 0} = \frac{2}{2} = 1$$

Both limits confirm that the function approaches $y=1$ as $x$ tends towards both positive and negative infinity. This reinforces our earlier conclusion based on comparing degrees.

Conclusion and Options

Based on our analysis, the function f(x)=rac{2 x^2+2}{2 x^2-2} has a horizontal asymptote at $y=1$. This is because the degrees of the numerator and the denominator are equal, and the ratio of their leading coefficients is rac{2}{2} = 1. The limit evaluation method also confirms this result.

Now let's look at the given options:

A. $y=-2$ B. $y=1$ C. $y=2$ D. None

Our calculated horizontal asymptote is $y=1$, which matches option B.

For further exploration into the fascinating world of functions and their graphical properties, you can visit resources like Khan Academy or Paul's Online Math Notes to deepen your understanding.