Slant asymptote of f(x) = (x^2 + 1)/(x - 1) Easy

The question

Question: find the slant asymptote of f(x) = (x² + 1)/(x − 1).

An asymptote is a line the graph gets closer and closer to.

We draw it dashed.

1 Compare the powers

The top has x²: power 2.

The bottom has x: power 1.

The top power is one bigger.

Then there is no flat asymptote.

Instead, the graph gets close to a slanted line: a slant asymptote.

Dividing finds it.

2 Set it up

We write it like long division with numbers.

x − 1 goes outside, x² + 1 inside.

There is no x part.

We write 0x, so the columns line up.

3 First part of the answer

Divide the first parts: x² divided by x is x.

We write it on top.

Multiply: x times (x − 1) is x² − x.

We write it under, with a minus: we take it away.

x² take away x² is 0.

That is why we chose x.

0x take away −x is x.

Bring down the next part: 1.

4 Second part of the answer

Divide the first parts: x divided by x is 1.

We write it on top.

Multiply: 1 times (x − 1) is x − 1.

We write it under, with a minus: we take it away.

x take away x is 0.

That is why we chose 1.

1 take away −1 is 2.

2 has no x.

We can't divide it by x.

So 2 is the remainder: what is left over.

So f(x) is x + 1, plus 2/(x − 1).

5 Read the slant asymptote

When x is big, 2/(x − 1) is tiny.

At x = 100 it is 2/99: about 0.02.

Almost 0.

So for a big x, f(x) is close to x + 1.

The slant asymptote is y = x + 1.

Careful!

The remainder part is not in the line.

It gets smaller and smaller, so we leave it out.

6 Vertical asymptote

The bottom: x − 1 = 0 when x = 1.

We can never divide by 0.

The top is not 0 there, so near it the graph shoots up or down: a vertical asymptote.

So the asymptotes are y = x + 1 and x = 1.

7 Check

Let's check with a picture.

The vertical asymptote x = 1, dashed.

The slant asymptote y = x + 1, dashed.

The graph of f: on each side it gets close to both dashed lines.

And with numbers: at x = 100, f is about 101.02, and the line gives 101.

Very close.

It works!

Let's look back.

First: compare the powers.

Then: set it up.

Then: first part of the answer.

Then: second part of the answer.

Then: read the slant asymptote.

Then: vertical asymptote.

Last: check.

▶ Watch it solved

Asymptotes of f(x) = (x^2 - x - 6)/(x - 1) Medium

The question

Question: find the asymptotes of f(x) = (x² − x − 6)/(x − 1).

An asymptote is a line the graph gets closer and closer to.

We draw it dashed.

1 Compare the powers

The top has x²: power 2.

The bottom has x: power 1.

The top power is one bigger.

Then there is no flat asymptote.

Instead, the graph gets close to a slanted line: a slant asymptote.

Dividing finds it.

2 Set it up

We write it like long division with numbers.

x − 1 goes outside, x² − x − 6 inside.

3 First part of the answer

Divide the first parts: x² divided by x is x.

We write it on top.

Multiply: x times (x − 1) is x² − x.

We write it under, with a minus: we take it away.

x² take away x² is 0.

That is why we chose x.

−x take away −x is 0x.

Bring down the next part: −6.

4 Second part of the answer

Divide the first parts: 0x divided by x is 0.

We write it on top.

Multiply: 0 times (x − 1) is 0.

We write it under, with a minus: we take it away.

0x take away 0x is 0.

That is why we chose 0.

−6 take away 0 is −6.

−6 has no x.

We can't divide it by x.

So −6 is the remainder: what is left over.

So f(x) is x, minus 6/(x − 1).

5 Read the slant asymptote

When x is big, 6/(x − 1) is tiny.

At x = 100 it is 6/99: about 0.06.

Almost 0.

So for a big x, f(x) is close to x.

The slant asymptote is y = x.

Careful!

The remainder part is not in the line.

It gets smaller and smaller, so we leave it out.

6 Vertical asymptote

The bottom: x − 1 = 0 when x = 1.

We can never divide by 0.

The top is not 0 there, so near it the graph shoots up or down: a vertical asymptote.

So the asymptotes are y = x and x = 1.

7 Check

Let's check with a picture.

The vertical asymptote x = 1, dashed.

The slant asymptote y = x, dashed.

The graph of f: on each side it gets close to both dashed lines.

And with numbers: at x = 100, f is about 99.94, and the line gives 100.

Very close.

It works!

Let's look back.

First: compare the powers.

Then: set it up.

Then: first part of the answer.

Then: second part of the answer.

Then: read the slant asymptote.

Then: vertical asymptote.

Last: check.

▶ Watch it solved

Asymptotes of f(x) = (3x^2 + 2x - 1)/(x + 2) Harder

The question

Question: find the asymptotes of f(x) = (3x² + 2x − 1)/(x + 2).

An asymptote is a line the graph gets closer and closer to.

We draw it dashed.

1 Compare the powers

The top has x²: power 2.

The bottom has x: power 1.

The top power is one bigger.

Then there is no flat asymptote.

Instead, the graph gets close to a slanted line: a slant asymptote.

Dividing finds it.

2 Set it up

We write it like long division with numbers.

x + 2 goes outside, 3x² + 2x − 1 inside.

3 First part of the answer

Divide the first parts: 3x² divided by x is 3x.

We write it on top.

Multiply: 3x times (x + 2) is 3x² + 6x.

We write it under, with a minus: we take it away.

3x² take away 3x² is 0.

That is why we chose 3x.

2x take away 6x is −4x.

Bring down the next part: −1.

4 Second part of the answer

Divide the first parts: −4x divided by x is −4.

We write it on top.

Multiply: −4 times (x + 2) is −4x − 8.

We write it under, with a minus: we take it away.

−4x take away −4x is 0.

That is why we chose −4.

−1 take away −8 is 7.

7 has no x.

We can't divide it by x.

So 7 is the remainder: what is left over.

So f(x) is 3x − 4, plus 7/(x + 2).

5 Read the slant asymptote

When x is big, 7/(x + 2) is tiny.

At x = 100 it is 7/102: about 0.07.

Almost 0.

So for a big x, f(x) is close to 3x − 4.

The slant asymptote is y = 3x − 4.

Careful!

The remainder part is not in the line.

It gets smaller and smaller, so we leave it out.

6 Vertical asymptote

The bottom: x + 2 = 0 when x = −2.

We can never divide by 0.

The top is not 0 there, so near it the graph shoots up or down: a vertical asymptote.

So the asymptotes are y = 3x − 4 and x = −2.

7 Check

Let's check with a picture.

The vertical asymptote x = −2, dashed.

The slant asymptote y = 3x − 4, dashed.

The graph of f: on each side it gets close to both dashed lines.

And with numbers: at x = 100, f is about 296.07, and the line gives 296.

Very close.

It works!

Let's look back.

First: compare the powers.

Then: set it up.

Then: first part of the answer.

Then: second part of the answer.

Then: read the slant asymptote.

Then: vertical asymptote.

Last: check.

▶ Watch it solved