Question

The graph of has a vertical asymptote at x =
.

Answer 1

It looks like the function is missing from your question. Could you paste the function or full problem?

Meanwhile, here’s how to find vertical asymptotes so you can apply it once you give the function:

1. If it’s a rational function f(x) = N(x)/D(x), find the zeros of D(x) (solve D(x) = 0). These are candidate vertical asymptotes.
2. Factor N and D and cancel any common factors. If a zero of D(x) cancels with a zero of N(x), that x-value is a removable discontinuity (a hole), not a vertical asymptote.
3. For any remaining zeros of D(x) (not canceled), check the one-sided limits:
- If lim_{x→a+} f(x) = ±∞ or lim_{x→a-} f(x) = ±∞, then x = a is a vertical asymptote.
4. Note multiplicity effects: if the factor (x − a) in the denominator has odd multiplicity the signs on either side of the asymptote will differ; if even multiplicity the signs will be the same.

Examples:
- f(x) = (x+1)/(x−2) → denominator zero at x=2 (not canceled) → vertical asymptote x = 2.
- f(x) = (x^2−1)/(x−1) = (x−1)(x+1)/(x−1) → cancels to x+1 → x = 1 is a hole, not an asymptote.
- g(x) = tan x → vertical asymptotes at x = π/2 + kπ, k ∈ Z.

Send the function and I’ll find the vertical asymptote(s) for you.