Question

For the function f whose graph is shown, state the following.

(a)\(\mathop {{\bf{lim}}}\limits_{x \to - {\bf{7}}} f\left( x \right)\)

(b)\(\mathop {{\bf{lim}}}\limits_{x \to - {\bf{3}}} f\left( x \right)\)

(c)\(\mathop {{\bf{lim}}}\limits_{x \to {\bf{0}}} f\left( x \right)\)

(d)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{6}}^ - }} f\left( x \right)\)

(e)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{6}}^ + }} f\left( x \right)\)

(f) The equation of vertical asymptotes

Short answer

(a)\( - \infty \)

(b) \(\infty \)

(c) \(\infty \)

(d) \( - \infty \)

(e)\(x = - 7\), \(x = - 3\), \(x = 0\), and \(x = 6\)

Step-by-step solution

Step 1:Find the value of part (a)

From the graph, it can be observed that the value of \(\mathop {\lim }\limits_{x \to - 7} f\left( x \right)\)(when x is approaching to \( - 7\)) is \( - \infty \).

Find the value of part (b)

From the graph, it can be observed that the value of \(\mathop {\lim }\limits_{x \to - 3} f\left( x \right)\)(when x is approaching to \( - 3\)) is \(\infty \).

Find the value of part (c)

From the graph, it can be observed that the value of \(\mathop {\lim }\limits_{x \to 0} f\left( x \right)\)(when x is approaching to 0) is \(\infty \).

Find the value of part (d)

From the graph, as the curve is approaching 6 from the left-hand side, the value of \(\mathop {\lim }\limits_{x \to {6^ - }} f\left( x \right)\) is \( - \infty \).

Find the value of part (e)

From the graph, as the curve is approaching 6 from the right-hand side, the value of \(\mathop {\lim }\limits_{x \to {6^ - }} f\left( x \right)\) is \(\infty \).

Find the value of part (f)

From the graph, the equations of the vertical asymptotes are \(x = - 7\), \(x = - 3\), \(x = 0\), and \(x = 6\).