# How do you find the vertical, horizontal or slant asymptotes for #(2x)/(x^2+16)#?

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (numerator-degree 1, denominator-degree 2) hence there are no slant asymptotes. graph{(2x)/(x^2+16) [-10, 10, -5, 5]}

By signing up, you agree to our Terms of Service and Privacy Policy

To find the vertical asymptotes of the function ( f(x) = \frac{2x}{x^2 + 16} ), we need to identify the values of ( x ) for which the denominator becomes zero. Setting the denominator equal to zero, we get:

[ x^2 + 16 = 0 ]

Solving this equation, we find that there are no real solutions, which means there are no vertical asymptotes.

To find the horizontal asymptote, we examine the behavior of the function as ( x ) approaches positive or negative infinity. As ( x ) becomes large, the terms involving ( x ) in the numerator and the denominator dominate the function. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ).

To determine if there are any slant asymptotes, we perform polynomial long division or use partial fraction decomposition to rewrite the function. After simplifying, if the resulting expression has a linear term that cannot be cancelled out, there will be a slant asymptote. In this case, performing polynomial long division:

[ \frac{2x}{x^2 + 16} = \frac{2x}{(x - 0)(x^2 + 16)} = \frac{2x}{x(x^2 + 16)} ]

Thus, there is no slant asymptote.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- If #(-1, 0)# lies on the graph of #y = f(x)#, what is the point that lies on the graph of #y = f(x + 3)#?
- How do you find the asymptotes for #4^(x-5)-5#?
- How do you use composition of functions to show that #f(x)=(2+x)/x# and #f^-1(x) = 2/(x-1)# are inverses?
- Consider the function #f(x)= 9x-x^3#. Is this function odd, even, or neither?
- How do you find the inverse of #y =1/logx#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7