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

### Examples

1. Find

### Show Answer

2. Find:

a)

b)

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a)

Since from the positive side when , hence, the '+' sign.

b)

3. Find .

### Show Answer

Dividing by will give , so divide by *x*.

## 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 and , given .

Horizontal and other asymptotes are found (if they exist) when finding .

- 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 .

### Show Answer

- Intercept:

For *x*-intercept, *y*=0

So the *x*-intercept is 0.

For *y*-intercept, *x*=0

So the *y*-intercept is 0.

- Asymptotes

Vertical asymptote:

So there is an asymptote at *x* = 2.

As *x* approaches 2 from LHS:

So *y* approaches infinity (negative).

As x approaches 2 from RHS:

So *y* approaches infinity (positive).

Horizontal asymptote:

This means that as *x* approaches , the function approaches *y* = *x*. As *x* approaches infinity,

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.

The point (1000, 1002) is just above the line *y* = *x*.

**Domain**: {*x*: all real }

**Range**:

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

When.

So the range is.

Putting all this information together gives the graph below.

Another method of solving inequations is solving them graphically.

2. Solve graphically.

Sketch and *y*=1 on the same number plane.

The solution of occurs when the hyperbola is below the line *y* = 1.