Answer:
A
Explanation:
an asymptote is a tangent to the function out in infinity (+ or -, x or y). a line that stands for the limit the function tends toward but will never reach (hence "infinity").
A is the right answer, because x²+16 never turns 0, so there is no break to "infinity" anywhere, and the function is continuous. and both numerator and denominator are polynomials of 2nd degree. so, for larger and larger x both get closer and closer to each other with the limit being x²/x² = 1.
so the horizontal asymptote is y = 1.
for a better understanding of the subject matter let's look at the other answer options, and why they are wrong.
B
it has the same limit 1 for larger and larger x, but because of x²-16 in the denominator it is not a continuous function.
at x = +4 and x = -4 this turns the denominator 0, which creates 2 undefined points and therefore 2 breaks in the function.
C
is ultimately a line function. it is a x² polynomial divided by a x polynomial, resulting in a x polynomial. a line has no asymptote.
D
this is kind of the opposite of C.
it is a x polynomial divided by a x² polynomial, resulting in a 1/x polynomial expression.
the limit of 1/x for larger and larger x is 0 and not 1.