How do you determine the vertical and horizontal asymptotes of the graph of each function # h(x) = (x+6)/(x^2 - 36)#?

Question

Lincoln Anderson · Accepted Answer

Vertical asymptotes occur at the points where the function tends to infinity (positive or negative). In case of this function these are points where the denominator equals to zero, that is #x=6# and #x=-6#. Horizontal asymptotes exist if the function tends to some constant as its argument tends to infinity (positive or negative). In this case, when #x->+-oo#, function tends to zero, so the X-axis is the horizontal asymptote for both directions of infinity for its argument.

James Allen · Answer

undefined To determine the vertical asymptotes of the function $ h(x) = \frac{x+6}{x^2 - 36} $, we need to find the values of $ x $ for which the denominator equals zero. Vertical asymptotes occur where the function is undefined, which happens when the denominator of the function is zero. Therefore, we set the denominator $ x^2 - 36 $ equal to zero and solve for $ x $:

$$ x^2 - 36 = 0 $$

$$ x^2 = 36 $$

$$ x = \pm 6 $$

So, the vertical asymptotes of the function occur at $ x = -6 $ and $ x = 6 $.

To determine the horizontal asymptote, we examine the behavior of the function as $ x $ approaches positive or negative infinity. We can do this by analyzing the degrees of the numerator and denominator polynomials. In this case, the degree of the numerator is 1, and the degree of the denominator is 2.

Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote will occur at $ y = 0 $ or the x-axis. Therefore, the horizontal asymptote of the function $ h(x) = \frac{x+6}{x^2 - 36} $ is the x-axis, represented by the equation $ y = 0 $.