** Definition**. The vertical line x = c is called a *vertical asymptote* to the graph of a function f if and only if either

or

or both.

** Definition**. The horizontal line y = L is a *horizontal asymptote* to the graph of a function f if and only if

or both.

** Definition**. When a linear asymptote is not parallel to the x- or y-axis, it is called an *oblique asymptote *or* slant asymptote*. A function f(x) is asymptotic to the straight line y = mx + q (m ≠ 0) if:

In the first case the line *y = mx + q* is an oblique asymptote of ƒ(x) when x tends to – ∞, and in the second case the line *y = mx + q *is an oblique asymptote of ƒ(x) when x tends to – ∞

The oblique asymptote, for the function f(x), will be given by the equation y=mx+n. The value for m is computed first and is given by the following limit:

It is good practice to treat the two cases (- ∞ and +∞) separately. If this limit doesn’t exist or is equal to zero, then there is no oblique asymptote in that direction.

Having m, then the value for q can be computed by

If this limit fails to exist then there is no oblique asymptote in that direction, even if a limit defining m exists.

**Exercise 1.** Find the vertical and horizontal asymptotes of the following function:

* Solution*. The domain is the set of all x-values that do not give a zero in the denominator. So we set the denominator equal to zero and solve the domain:

Since we can’t have a zero in the denominator, then we can’t have x = 3. Then, the domain is:

Now, we find the vertical asymptotes of the assigned function using the definition of vertical asymptote:

Then, the line x = 3 is the vertical asymptote.

Now, we find the horizontal asymptote of the assigned function using the definition of horizontal asymptote.

The line y = 4 is the horizontal asymptote.

**Exercise 2**. Find the asymptotes of the following function:

* Solution*. The domain is the set of all x-values that do not give a zero in the denominator. Since in this case there are no zeroes in the denominator, then there are no forbidden x-values, and the domain is:

Also, since there are no values forbidden to the domain, there are no vertical asymptotes.

Now, we find the horizontal asymptote of the assigned function using the definition of horizontal asymptote:

So, there is no horizontal asymptote. This result occurs when the degree in the numerator is greater than in the denominator. Now, we can use the definition to obtain the slant asymptote:

Then, the value for m is: m = 1.

Having m, then the value for q can be computed by the following limit.

Then, the value for q is: q = 0 and the line y = x the slant asymptote.

*1*. Functions and their graph. One–one and onto functions. Composite and inverse functions.

*2*. Linear inequalities. Systems of linear inequalities

*3*. Polynomial inequalities. Rational inequalities. Absolute-value inequalities

*6*. Differentiation. Derivative rules, the chain rule.

*7*. Application of differentiation: L'Hospital's Rule

*8*. Vertical, Horizontal and Slant asymptotes

*9*. Higher Order Derivatives. Applications of differentiation: local and absolute extremes of a function

*10*. Curve Sketching

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