# For what values of x, if any, does #f(x) = 1/((12x-9)sin(pi+(3pi)/x) # have vertical asymptotes?

We can split this into two parts:

Thus:

Rearranging:

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

The function f(x) has vertical asymptotes at values of x where the denominator of the function becomes zero. In this case, the denominator is (12x-9)sin(pi+(3pi)/x). To find the values of x that make the denominator zero, we set it equal to zero and solve for x. However, we need to be careful because the sine function has periodic behavior.

Setting the denominator equal to zero: (12x-9)sin(pi+(3pi)/x) = 0

To find the values of x that make the sine function zero, we have: sin(pi+(3pi)/x) = 0

The sine function is zero at integer multiples of pi. So, we can write: pi + (3pi)/x = n*pi, where n is an integer

Simplifying the equation: (3pi)/x = (n-1)*pi 3/x = n-1

Solving for x: x = 3/(n-1)

Therefore, the function f(x) has vertical asymptotes at x = 3/(n-1), where n is an integer except when n = 1.

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.

- How do you evaluate the limit #lim(x^2-36)/(x+6)dx# as #x->1#?
- How do you evaluate the limit #lim e^t/t# as #t->oo#?
- How do you evaluate the limit #(x+3)^1997# as x approaches #-4#?
- What is the limit of #sqrt(2-T) - (sqrt 2)/T# as T approaches 0?
- How do you use the formal definition of a limit to prove #lim(x^3 + 3x + 5) = 19# as x approaches 2?

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