Find the limit of the following functions as x goes to Infinity.
(a) f(x) = sin[ (x^2 + 4) / (x^2 - 4) ]
(b) f(x) = (4 sin x)/x
When looking for vertical asymptotes the first thing to look at is the denominator. If the denominator goes to 0 then you might have a vertical asymptote. Further, if the numerator does not go to 0 at the same points as the denominator, then you have a vertical asymptote. There are two of these here. Can you find them?
As for horizontal asymptotes you want to take the limit as x goes to either positive or negative infinity. As x gets large what happens to the numerator? To the denominator?
-Dan
x = -2 and x = 2 make the denominator equal to zero. When I look at the graph I see a very bold oscillation at these two points, but I don't see clearly the function increasing or decreasing without bound at them.
Can these two points be vertical asymptotes? What are their direction, positive infinity or negative infinity?
When I take the limit as x goes to infinity or negative infinity, I get the same answer which is sin(1).
sin(1) = y = horizontal asymptotes
no vertical asymptotes ... as $x \to \pm 2$ the argument of the sine function gets very large positive or negative causing an increasing rate of oscillation between $y=-1$ and $y=1$ as $x$ gets closer to $\pm 2$