Question

directions: write a limit statement describing the output values for the following graphs and verbal descriptions of the input values.

1. the input values decrease without bound
2. the input values increase without bound

graph of f(x)
graph of g(x)

1. limit statement:
2. limit statement:

directions: the graphs of the functions h and k are given below. use the graphs to find the following limits.
graph of h(x)

1. \\(\lim_{x\to2^{-}}h(x)=\\)
2. \\(\lim_{x\to2^{+}}h(x)=\\)
3. \\(\lim_{x\to3^{-}}h(x)=\\)
4. \\(\lim_{x\to3^{+}}h(x)=\\)

graph of k(x)

1. \\(\lim_{x\to5^{-}}k(x)=\\)
2. \\(\lim_{x\to3^{+}}k(x)=\\)
3. \\(\lim_{x\to -\infty}k(x)=\\)
4. \\(\lim_{x\to\infty}k(x)=\\)

review a: topics 1.7 - 1.11
rational functions
created by bryan passwater

Explanation:

Step1: Analyze graph of \(f(x)\) for decreasing input

As \(x\) decreases without bound (i.e., \(x
ightarrow-\infty\)) on the graph of \(f(x)\), the function value \(y = f(x)\) increases without bound. So the limit statement is \(\lim_{x
ightarrow-\infty}f(x)=\infty\).

Step2: Analyze graph of \(g(x)\) for increasing input

As \(x\) increases without bound (i.e., \(x
ightarrow\infty\)) on the graph of \(g(x)\), the function value \(y = g(x)\) approaches \(1\). So the limit statement is \(\lim_{x
ightarrow\infty}g(x)=1\).

Step3: Analyze left - hand limit of \(h(x)\) at \(x = 2\)

Looking at the graph of \(h(x)\), as \(x\) approaches \(2\) from the left (\(x
ightarrow2^{-}\)), the function value \(y=h(x)\) approaches \(-\infty\). So \(\lim_{x
ightarrow2^{-}}h(x)=-\infty\).

Step4: Analyze right - hand limit of \(h(x)\) at \(x = 2\)

As \(x\) approaches \(2\) from the right (\(x
ightarrow2^{+}\)) on the graph of \(h(x)\), the function value \(y = h(x)\) approaches \(\infty\). So \(\lim_{x
ightarrow2^{+}}h(x)=\infty\).

Step5: Analyze left - hand limit of \(h(x)\) at \(x = 3\)

As \(x\) approaches \(3\) from the left (\(x
ightarrow3^{-}\)) on the graph of \(h(x)\), the function value \(y=h(x)\) approaches \(3\). So \(\lim_{x
ightarrow3^{-}}h(x)=3\).

Step6: Analyze right - hand limit of \(h(x)\) at \(x = 3\)

As \(x\) approaches \(3\) from the right (\(x
ightarrow3^{+}\)) on the graph of \(h(x)\), the function value \(y = h(x)\) approaches \(3\). So \(\lim_{x
ightarrow3^{+}}h(x)=3\).

Step7: Analyze left - hand limit of \(k(x)\) at \(x = 5\)

As \(x\) approaches \(5\) from the left (\(x
ightarrow5^{-}\)) on the graph of \(k(x)\), the function value \(y=k(x)\) approaches \(-\infty\). So \(\lim_{x
ightarrow5^{-}}k(x)=-\infty\).

Step8: Analyze right - hand limit of \(k(x)\) at \(x = 3\)

As \(x\) approaches \(3\) from the right (\(x
ightarrow3^{+}\)) on the graph of \(k(x)\), the function value \(y = k(x)\) approaches \(-\infty\). So \(\lim_{x
ightarrow3^{+}}k(x)=-\infty\).

Step9: Analyze limit of \(k(x)\) as \(x

ightarrow-\infty\)
As \(x\) approaches \(-\infty\) on the graph of \(k(x)\), the function value \(y = k(x)\) approaches \(2\). So \(\lim_{x
ightarrow-\infty}k(x)=2\).

Step10: Analyze limit of \(k(x)\) as \(x

ightarrow\infty\)
As \(x\) approaches \(\infty\) on the graph of \(k(x)\), the function value \(y = k(x)\) approaches \(2\). So \(\lim_{x
ightarrow\infty}k(x)=2\).

Answer:

1. \(\lim_{x

ightarrow-\infty}f(x)=\infty\)

1. \(\lim_{x

ightarrow\infty}g(x)=1\)

1. \(\lim_{x

ightarrow2^{-}}h(x)=-\infty\)

1. \(\lim_{x

ightarrow2^{+}}h(x)=\infty\)

1. \(\lim_{x

ightarrow3^{-}}h(x)=3\)

1. \(\lim_{x

ightarrow3^{+}}h(x)=3\)

1. \(\lim_{x

ightarrow5^{-}}k(x)=-\infty\)

1. \(\lim_{x

ightarrow3^{+}}k(x)=-\infty\)

1. \(\lim_{x

ightarrow-\infty}k(x)=2\)

1. \(\lim_{x

ightarrow\infty}k(x)=2\)