Question

The graph of has a vertical asymptote at x =

Answer 1

It looks like part of the problem is missing — I don’t know which function or graph you mean. Could you paste the function or an image of the graph?

Meanwhile, here’s how to find vertical asymptotes in common cases:

- For a rational function f(x) = P(x)/Q(x):
1. Factor P and Q and simplify (cancel any common factors).
2. Solve Q(x) = 0. Each solution x = a that remains after cancellation is a vertical asymptote.
3. If a factor cancels (P and Q both zero at x = a), that x is a removable discontinuity (a hole), not a vertical asymptote.
4. Multiplicity matters: if (x−a) has odd multiplicity in Q, the graph changes sign across the asymptote; if even, it stays on the same side.

- For logarithmic functions, ln(x−a) has a vertical asymptote at x = a (where the log argument is zero).

- For other types, look for x-values where the function goes to ±∞ (denominator → 0 without cancellation, or domain boundary like log).

If you give the function or graph, I’ll find the exact vertical asymptote(s).