Question

Find the vertical and horizontal asymptotes. Write the asymptotes as equations of lines. \(f(x)=\frac{2+x}{1-x}\)

Short answer

The vertical asymptote for the function \(f(x)=\frac{2+x}{1-x}\) is \(x = 1\) and the horizontal asymptote (for both \(x \rightarrow +\infty\) and \(x \rightarrow -\infty\)) is \(y = -1\).

Step-by-step solution

Finding Vertical Asymptotes

Set the denominator of the function equal to zero and solve for x: \(1-x = 0\). Solving this equation gives \(x = 1\). So the vertical asymptote is \(x = 1\). This is a line that the graph of the function will tend to but never actually touch or cross.

Finding Horizontal Asymptotes for \(x \rightarrow +\infty\)

Take the limit of the function as \(x\) approaches plus infinity: \(\lim_{x \rightarrow +\infty} f(x) = \lim_{x \rightarrow +\infty} \frac{2+x}{1-x}\). By looking at the highest powers in the numerator and denominator, this limit is equal to \(-1\). So, one of the horizontal asymptotes is \(y = -1\). This is a line that the graph of the function will tend to but never reach as \(x\) approaches positive infinity.

Finding Horizontal Asymptotes for \(x \rightarrow -\infty\)

Take the limit of the function as \(x\) approaches minus infinity: \(\lim_{x \rightarrow -\infty} f(x) = \lim_{x \rightarrow -\infty} \frac{2+x}{1-x}\). By looking at the highest powers in the numerator and denominator, this limit is also equal to \(-1\). So, the other horizontal asymptote is also \(y = -1\). This is a line that the graph of the function will tend to but never reach as \(x\) approaches negative infinity.

Key concepts

Vertical Asymptotes

Vertical asymptotes occur in a graph where the function's value becomes infinitely large or infinitely small as it gets closer to a certain value of 'x'. Imagine it as an invisible boundary that the function approaches but never crosses or touches. This situation mainly arises in rational functions, where the denominator of the function equals zero, creating a division by zero error, which is undefined.

For instance, in the function \(f(x)=\frac{2+x}{1-x}\), we examine the denominator to find the vertical asymptote. The equation \(1-x = 0\) yields \(x = 1\) after solving it. Hence, the vertical asymptote for this function is \(x = 1\). Graphically, this is a 'line' parallel to the y-axis that the graph will infinitely approach as \(x\) nears 1.

Horizontal Asymptotes

Unlike vertical asymptotes, horizontal asymptotes represent the value that a function approaches as \(x\) wends its way towards infinity or negative infinity. These asymptotes give us a visual depiction of the function's end-behaviour.

In the case of our textbook exercise for the function \(f(x)=\frac{2+x}{1-x}\), we find the horizontal asymptote by taking the limit as \(x\) goes to positive or negative infinity. The function simplifies to \(y = -1\) in both directions of the infinity, suggesting that the function will draw closer to \(y = -1\) but will not actually ever reach this value. Therefore, \(y = -1\) is the horizontal asymptote for this function, acting as a boundary line parallel to the x-axis.

Limit of a Function

The \emph{limit\} of a function is a fundamental concept in calculus that describes the behaviour of a function as it approaches a certain point. It's like getting as close as possible to a destination without necessarily reaching it. In the context of asymptotes, limits help determine the line that the function will approach but not touch.

Specifically, limits like \(\lim_{x \rightarrow \pm\infty} f(x)\) can identify horizontal asymptotes. If the limit exists and is a finite number as \(x\) approaches infinity, then that number is a horizontal asymptote. For \(f(x)=\frac{2+x}{1-x}\), we calculated the limit as \(x\) approaches both positive and negative infinity and consistently found it to be \(y = -1\), giving us the horizontal asymptote. Understanding limits is critical to graphing and analyzing rational functions specially.

Rational Functions

Rational functions are ratios of two polynomials. They are in the form \(f(x)=\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomial functions, and \(Q(x)\) is not the zero polynomial. Due to the nature of division by a polynomial, rational functions can have both vertical and horizontal asymptotes, displaying unique behaviours as \(x\) approaches specific values.

The beauty of rational functions lies in their complexity and predictability. By identifying zeroes of the denominator, we can locate vertical asymptotes. Similarly, comparing the degrees of the numerator and denominator polynomials can reveal horizontal asymptotes. As in our example \(f(x)=\frac{2+x}{1-x}\), a careful analysis shows that not only does the function have both vertical and horizontal asymptotes, but also that the asymptotes extensively dictate the shape and behaviour of its graph.