Asymptotes of any curve. Hello friends, today it’s all about the asymptotes of any curve. Have a look!!

**Asymptotes of any curve**

Suppose is the equation of any curve. Then for the asymptote of this curve parallel to the -axis, the coefficient of the highest power of will be zero.

Similarly, for the asymptote of this curve parallel to the -axis, the coefficient of the highest power of will be zero.

For the general equation of an asymptote, first I’ll assume . Then I’ll substitute this value at the equation of the curve. Next, I’ll equate the coefficients of the two highest powers of to zero. And that will give the general equation of an asymptote.

Now I will give some examples.

**Examples of the asymptotes of any curve**

Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

**Example 1**

According to Stroud and Booth (2013)* “For each of the following curves, determine the asymptotes parallel to the and axes: .”

##### Solution

Now here the given curve is

Next, I’ll rewrite it as

Ok, so for any asymptote parallel to the axis, the coefficient of the highest power of will be zero.

Now here is the highest power of . And the coefficient of is .

Thus is not possible.

Therefore there is no asymptote of this curve parallel to the axis.

Next, I’ll check if this curve has an asymptote parallel to the axis.

So, for any asymptote parallel to the axis, the coefficient of the highest power of will be zero.

Now here is the highest power of . And the coefficient of is .

Thus I can say . And this gives

which is the asymptote of this curve parallel to the axis.

Hence I can conclude that the curve has no asymptote parallel to the axis. Also, is the asymptote of this curve parallel to the axis. And these are the answers to the given example.

Now I’ll give another example.

**Example 2**

According to Stroud and Booth (2013)* “For each of the following curves, determine the asymptotes parallel to the and axes: .”

**Solution**

Now here the given curve is

(1)

which gives

(2)

Ok, so for any asymptote parallel to the axis, the coefficient of the highest power of will be zero. Now here is the highest power of . And, from equation (2), I can say that the coefficient of is . Thus I can say .

And this gives

So I get

which are the asymptotes of this curve parallel to the axis. Next, I’ll check if this curve has an asymptote parallel to the axis.

So, for any asymptote parallel to the axis, the coefficient of the highest power of will be zero. Now here is the highest power of . And, from equation (1), I can say that the coefficient of is .

Thus I can say . And this gives

which is the asymptote of this curve parallel to the axis.

Hence I can conclude that are the asymptotes of this curve parallel to the axis. Also, is the asymptote of this curve parallel to the axis. And these are the answers to the given example.

Next comes my other example.

##### Example 3

According to Stroud and Booth (2013)* “Determine all the asymptotes of each of the following curves: .”

**Solution**

(3)

And I have to get all the asymptotes of this curve. So I’ll start with the asymptotes parallel to the – and – axes.

**Step 1**

As in examples 1 and 2, for any asymptote parallel to the axis, the coefficient of the highest power of will be zero.

Now here is the highest power of . And, from equation (3), I can say that the coefficient of is .

Thus is the asymptote of this curve parallel to the axis. Next, I’ll check if this curve has an asymptote parallel to the axis. So, for any asymptote parallel to the axis, the coefficient of the highest power of will be zero.

Now here is the highest power of . And, from equation (3), I can say that the coefficient of is . Thus is not possible. Therefore there is no asymptote of this curve parallel to the axis.

Next, I’ll get the general asymptotes of the curve in equation (3).

**Step 2**

Now, for the general equation of an asymptote, let me assume

(4)

And I’ll substitute this in equation (3) to get

Next, I’ll simplify it by using the formulas and . So this gives

Then I’ll bring together the coefficients of and constants to get

Now I have to equate the coefficients of the two highest powers of to . So here the two highest powers of are and . And the coefficient of is . So this means

(5)

Again the coefficient of is . And that gives

(6)

Next, I’ll solve equations (5) and (6) to get the values of and .

Now I can rewrite equation (5) as

So this means

And this gives

Next, I’ll substitute these values of in the equation (6) to get the respective values of .

**Step 3**

So I’ll put in the equation (6) to get

Thus I can say for . Now I’ll substitute these two values in the equation (4) to get

Hence is one of the general asymptotes. But is the asymptote of this curve parallel to the axis. So I’ll discard as one of the general asymptotes.

Now I’ll put in the equation (6) to get

So this gives

Thus I can say for . Now I’ll substitute these two values in the equation (4) to get

Hence is the second general asymptote.

Next, I’ll put in the equation (6) to get

So this gives

Thus I can say for . Now I’ll substitute these two values in the equation (4) to get

Therefore is the third general asymptote.

Hence I can conclude that is the asymptote of this curve parallel to the axis. Also, are the two general asymptotes of this curve. And this is the answer to the given example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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