Limits, how do I find the horizontal asymptotes?

Question

For f(x)= (sqrt36x⁴+7)/(9x²+4) 

 

I am really lost

Answer 1

Horizontal asymptotes are invisible lines that the graph of the function approach but never touch.

So the horizontal asymptote is the limit of f(x) as x --> ± infinity

 

Method; Step one:

 

evaluate/compare degree's of x in the numerator and denominator polynomials.

 

Numerator: 2nd degree polynomial

* going to assume √(36x⁴+7) since there are no parenthesis

 

Square root is 1/2 degree

 

(x⁴)^1/2 = x² = 2nd degree polynomial 

 

Denominator: x² = 2nd degree polynomial

 

If the degrees are the same, which they are in this case.

 

Step two:

 

Look at the coefficients of the numerator and denominator

 

Numerator:

 

√36 = 6

 

Denominator:

 

9

 

6/9 = 2/3

 

y = 2/3 is the horizontal asymptote

 

Then the limit;

 

Lim (sqrt36x⁴+7)/(9x²+4) = 2/3

x --> infinity

Answer 2

To find the horizontal asymptotes of a function f(x), we need to determine how f(x) behaves as x approaches ∞ or -∞.

 

lim_x→∞ [√(36x⁴+7)/(9x²+4)] = lim_x→∞ [√x⁴ √(36+7/x⁴)/(9x²+4)]

 

                                        = lim_x→∞ [x² √ (36 + 7/x⁴)/(9x²+4)]

 

                                        = lim_x→∞ [√(36+7/x⁴)/(9+4/x²)

 

                                        = √(36+0)/(9+0) = 6/9 = 2/3

 

We get the same value for the limit of the function as x → -∞.

 

So, the only horizontal asymptote is y = 2/3.