Question

For the function g whose graph is given, state the following.

(a) \(\mathop {{\bf{lim}}}\limits_{x \to \infty } \,\,g\left( x \right)\) (b) \(\mathop {{\bf{lim}}}\limits_{x \to - \infty } \,g\left( x \right)\) (c) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{0}}} \,\,g\left( x \right)\) (d) \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{2}}^ - }} \,\,g\left( x \right)\) (e) \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{2}}^ + }} \,\,g\left( x \right)\)

(f) The equations of asymptotes

Short answer

(a) 2

(b) \( - 1\)

(c) \( - \infty \)

(d) \( - \infty \)

(e) \(\infty \)

(e) \(x = 0\) and \(x = 2\) are the vertical asymptote and \(y = - 1\) and \(y = 2\) are horizontal asymptote.

Step-by-step solution

Find an answer for part (a)

From the graph, the value of the expression \(\mathop {\lim }\limits_{x \to \infty } g\left( x \right)\) (as xapproaching to \(\infty \)) is:

\(\mathop {\lim }\limits_{x \to \infty } g\left( x \right) = 2\)

Find an answer for part (b)

From the graph, the value of the expression \(\mathop {\lim }\limits_{x \to - \infty } g\left( x \right)\) (as x approaching to \( - \infty \)) is:

\(\mathop {\lim }\limits_{x \to - \infty } g\left( x \right) = - 1\)

Find an answer for part (c)

From the graph, the value of the expression \(\mathop {\lim }\limits_{x \to 0} g\left( x \right)\) (as x approaches to 0) is:

\(\mathop {\lim }\limits_{x \to 0} g\left( x \right) = - \infty \)

Find an answer for part (d)

From the graph, the value of the expression \(\mathop {\lim }\limits_{x \to {2^ - }} g\left( x \right)\) (as x approaching to 2 from the left) is:

\(\mathop {\lim }\limits_{x \to {2^ - }} g\left( x \right) = - \infty \)

Find the answer for part (e)

From the graph, the value of the expression \(\mathop {\lim }\limits_{x \to {2^ + }} g\left( x \right)\) (as x approaching to 2 from the right) is:

\(\mathop {\lim }\limits_{x \to {2^ + }} g\left( x \right) = \infty \)

Find the answer for part (f)

From the graph, it can be observed that:

(1) \(x = 0\) and \(x = 2\) are the vertical asymptote.

(2) \(y = - 1\) and \(y = 2\) are thehorizontal asymptote.