Question

Question 47-52: Find the horizontal and vertical asymptotes of each curve. You may want to use a graphing calculator (or computer) to check your work by graphing the curve and estimating the asymptotes.

47. \(y = \frac{{5 + 4x}}{{x + 3}}\)

Short answer

The line\(y = 4\)is a horizontal asymptote of the curve. The vertical line\(x = - 3\)is the vertical asymptotes of the curve.

Step-by-step solution

Condition for horizontal asymptotes and vertical asymptotes

The horizontal asymptote of the curve \(y = f\left( x \right)\) for the line \(y = L\) is\(\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = L\) or \(\mathop {\lim }\limits_{x \to - \infty } f\left( x \right) = L\).

The vertical asymptote of the curve\(y = f\left( x \right)\)for the line \(x = a\)are shown below:

\(\begin{aligned}{l}\mathop {\lim }\limits_{x \to a} f\left( x \right) = \infty \,\,\,\,\,\,\,\,\,\mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = \infty \,\,\,\,\,\,\,\,\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = \infty \\\mathop {\lim }\limits_{x \to a} f\left( x \right) = \infty \,\,\,\,\,\,\,\,\,\,\mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = - \infty \,\,\,\,\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = - \infty \end{aligned}\)

Determine the horizontal and vertical asymptotes of the curve

Divide both numerator and denominator by the highest power of\(x\)in the denominator.

\(\begin{aligned}\mathop {\lim }\limits_{x \to \pm \infty } \frac{{5 + 4x}}{{x + 3}} &= \mathop {\lim }\limits_{x \to \pm \infty } \frac{{\frac{{\left( {5 + 4x} \right)}}{x}}}{{\frac{{\left( {x + 3} \right)}}{x}}}\\ &= \mathop {\lim }\limits_{x \to \pm \infty } \frac{{\frac{5}{x} + 4}}{{1 + \frac{3}{x}}}\\ &= \frac{{0 + 4}}{{1 + 0}}\\ &= 4\end{aligned}\)

Therefore, the line\(y = 4\)is the horizontal asymptote of the curve.

Write the function as\(y = f\left( x \right) &= \frac{{5 + 4x}}{{x + 3}}\).

Determine the vertical asymptotes of the curve as shown below:

\(\mathop {\lim }\limits_{x \to - {3^ + }} f\left( x \right) = - \infty \,\,\,\,\left( {{\mathop{\rm since}\nolimits} \,\,5 + 4x \to - 7\,\,\,{\mathop{\rm and}\nolimits} \,\,x + 3 \to {0^ + }\,\,{\mathop{\rm as}\nolimits} \,\,x \to - {3^ + }} \right)\)

Therefore, the vertical line \(x = - 3\) is the vertical asymptotes of the curve.

Check the answer by graphing the curve 

The procedure to draw the graph of the equation by using the graphing calculator is as follows:

To check the answer, draw the graph of the function\(f\left( x \right) = \frac{{5 + 4x}}{{x + 3}}\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\left( {5 + 4X} \right)/\left( {X + 3} \right)\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the function \(f\left( x \right) = \frac{{5 + 4x}}{{x + 3}}\) as shown below:

It is observed from the graph that the line \(y = 4\) is the horizontal asymptote and \(x = - 3\) is the vertical asymptote.