Further Graphs

Graphs of functions with asymptotes can be difficult to sketch. It is important to find the limits as the function approaches the asymptotes.

A special limit is $\lim _{ x\rightarrow \infty }{ \cfrac { 1 }{ x } } =0$

Examples

1. Find $\lim _{ x\rightarrow \infty }{ \cfrac { 3{ x }^{ 2 } }{ { x }^{ 2 }-2x+3 } }$ $\lim _{ x\rightarrow \infty }{ \cfrac { 3{ x }^{ 2 } }{ { x }^{ 2 }-2x+3 } } \\ \\ =\lim _{ x\rightarrow \infty }{ \cfrac { \cfrac { 3{ x }^{ 2 } }{ { x }^{ 2 } } }{ \cfrac { { x }^{ 2 } }{ { x }^{ 2 } } -\cfrac { 2x }{ { x }^{ 2 } } +\cfrac { 3 }{ { x }^{ 2 } } } } \quad (divide\quad by\quad the\quad highest\quad power\quad of\quad x)\\ \\ =\lim _{ x\rightarrow \infty }{ \cfrac { 3 }{ 1-\cfrac { 2 }{ { x } } +\cfrac { 3 }{ { x }^{ 2 } } } } \\ \\ =\cfrac { 3 }{ 1-0+0 } \\ \\ =\quad 3$

2. Find:

a) $\lim _{ x\rightarrow \infty }{ \cfrac { x }{ { x }^{ 2 }+4x+4 } }$

b) $\lim _{ x\rightarrow - \infty }{ \cfrac { x }{ { x }^{ 2 }+4x+4 } }$

a) $\lim _{ x\rightarrow \infty }{ \cfrac { x }{ { x }^{ 2 }+4x+4 } } \\ \\ =\lim _{ x\rightarrow \infty }{ \cfrac { \cfrac { x }{ { x }^{ 2 } } }{ \cfrac { { x }^{ 2 } }{ { x }^{ 2 } } +\cfrac { 4x }{ { x }^{ 2 } } +\cfrac { 4 }{ { x }^{ 2 } } } } \\ \\ =\lim _{ x\rightarrow \infty }{ \cfrac { \cfrac { 1 }{ x } }{ 1-\cfrac { 4 }{ { x } } +\cfrac { 4 }{ { x }^{ 2 } } } } \\ \\ =\cfrac { 0 }{ 1+0+0 } \\ \\ =\quad { 0 }^{ + }$

Since $\cfrac{1}{x} \rightarrow 0$ from the positive side when $x \rightarrow + \infty$, hence, the '+' sign.

b) $\lim _{ x\rightarrow -\infty }{ \cfrac { x }{ { x }^{ 2 }+4x+4 } } \\ \\ =\lim _{ x\rightarrow -\infty }{ \cfrac { \cfrac { x }{ { x }^{ 2 } } }{ \cfrac { { x }^{ 2 } }{ { x }^{ 2 } } +\cfrac { 4x }{ { x }^{ 2 } } +\cfrac { 4 }{ { x }^{ 2 } } } } \\ \\ =\lim _{ x\rightarrow -\infty }{ \cfrac { \cfrac { 1 }{ x } }{ 1-\cfrac { 4 }{ { x } } +\cfrac { 4 }{ { x }^{ 2 } } } } \\ \\ =\cfrac { 0 }{ 1+0+0 } \\ \\ =\quad { 0 }^{ - }$

3. Find $\lim _{ x\rightarrow \infty }{ \cfrac { 3{ x }^{ 2 } }{ x-1 } }$.

Dividing by ${x}^{2}$ will give $\cfrac{3}{0}$, so divide by x. $\lim _{ x\rightarrow \infty }{ \cfrac { 3{ x }^{ 2 } }{ x-1 } } \\ \\ =\lim _{ x\rightarrow \infty }{ \cfrac { \cfrac { 3{ x }^{ 2 } }{ x } }{ \cfrac { x }{ x } -\cfrac { 1 }{ x } } } \\ \\ =\lim _{ x\rightarrow \infty }{ \cfrac { 3x }{ 1-\cfrac { 1 }{ x } } } \\ \\ =\cfrac { 3x }{ 1-0 } \\ \\ =3x$

General Graphs

A number of features are important in drawing graph:

• Intercepts

The x-intercept occurs when y=0.

The y-intercept occurs when x=0.

• Asymptotes

Vertical asymptotes occur when $f(x) \neq 0$ and $h(x)=0$, given $f(x)=\cfrac{g(x)}{h(x)}$.

Horizontal and other asymptotes are found (if they exist) when finding $\lim _{ x\rightarrow \pm \infty }{ f(x) }$.

• Domain and Range

The domain is the set of all possible x values for a function.

The range is the set of all possible y values for a function.

Examples

1. Sketch $f(x)=\cfrac{{x}^{2}}{x-2}$.

• Intercept:

For x-intercept, y=0 $\cfrac { { x }^{ 2 } }{ x-2 } =0\\ \\ { x }^{ 2 }=0\\ \\ x=0$

So the x-intercept is 0.

For y-intercept, x=0 $y=\cfrac { { 0 }^{ 2 } }{ 0-2 } \\ \\ y=0$

So the y-intercept is 0.

• Asymptotes

Vertical asymptote: $x-2\neq 0\\ x\neq 2$

So there is an asymptote at x = 2.

As x approaches 2 from LHS: $f({ 2 }^{ - })=\cfrac { { ({ 2 }^{ - }) }^{ 2 } }{ { 2 }^{ - }-2 } \\ \\ \qquad \quad =\quad \cfrac { + }{ - } \\ \\ \qquad \quad =\quad -$

So y approaches infinity (negative).

As x approaches 2 from RHS: $f({ 2 }^{ + })=\cfrac { { ({ 2 }^{ + }) }^{ 2 } }{ { 2 }^{ + }-2 } \\ \\ \qquad \quad =\quad \cfrac { + }{ + } \\ \\ \qquad \quad =\quad +$

So y approaches infinity (positive).

Horizontal asymptote: $\lim _{ x\rightarrow \infty }{ \cfrac { { x }^{ 2 } }{ x-2 } } \\ \\ =\lim _{ x\rightarrow \infty }{ \cfrac { \cfrac { { x }^{ 2 } }{ x } }{ \cfrac { x }{ x } -\cfrac { 2 }{ x } } } \\ \\ =\lim _{ x\rightarrow \infty }{ \cfrac { { x } }{ 1-\cfrac { 2 }{ x } } } \\ \\ =\cfrac { x }{ 1-0 } \\ \\ =\quad x$

This means that as x approaches $\pm \infty$, the function approaches yx. As x approaches infinity, $f(\infty )=\cfrac { { \infty }^{ 2 } }{ \infty -2 } >x\\ \\ So\quad as\quad x\rightarrow \infty ,\quad y\rightarrow x\quad from\quad above\\ As\quad x\rightarrow \infty \\ \\ f(-\infty )=\cfrac { { (-\infty ) }^{ 2 } }{ -\infty -2 }

This is not easy to see, so substitute extreme values such as 1000 and -1000 to see what y does as x approaches infinity on both sides. $f(1000)=\cfrac { { (1000) }^{ 2 } }{ 1000-2 } \\ \qquad \qquad \approx \quad 1002$

The point (1000, 1002) is just above the line yx.

Domain: {x: all real $x \neq 2$}

Range:

When x > 2, we find that an approximate range is y > 35 (substituting different values of x).

When $x < 2, y \le 0$.

So the range is ${y: y > 35, y \le 0}$.

Putting all this information together gives the graph below. Another method of solving inequations is solving them graphically.

2. Solve $\cfrac{1}{x-2} < 1$ graphically.

Sketch $y=\cfrac { 1 }{ x-2 } <1$ and y=1 on the same number plane. The solution of $\cfrac{1}{x-2} < 1$ occurs when the hyperbola $y=\cfrac { 1 }{ x-2 }$ is below the line y = 1.