calculate limit

Question

A function is said to have a horizontal asymptote if either the limit at infinity exists or the limit at negative infinity exists.
Show that the following functions has a horizontal asymptote by calculating the given limit.

 

lim(x->infinity) (14-3x)/(2+x) + ((14x^2)+(15))/(15x-5)^2

 

Answer 1

lim            14 - 3x       +        14x² + 15

x→∞        _______                ____________

                   2 + x                     25(3x - 1)²

 

 

You first realize that when you plug in x=∞, you get an indeterminate form.  So you must rewrite the limit to eliminate that possibility.  Since you are adding rationals, find the LCD and write as a single rational.  LCD is 25(2 + x)(3x - 1)².

 

The numerator is then written as

 

25(14 - 3x)(3x - 1)² + (14x + 5)(2 + x)

 

 

Simplify this and divide it by the LCD.  From there, divide the numerator and denominator by the highest power of x in the denominator.  This will allow to use the concept 1/x=0  when  x→∞.  Then, plug in x=∞ to cancel out terms.  In the end, your limit should be a constant value.  If not, then a horizontal asymptote does not exist.