Question

Which statement describes the behavior of the graph of the function shown at the vertical asymptotes?

As x approaches -7 from the left, y approaches ∞.
As x approaches -7 from the right, y approaches -∞.
As x approaches 7 from the left, y approaches –∞.
As x approaches 7 from the right, y approaches –∞.

Answer 1

I don’t have the graph you mention, so I can’t pick which statement is true. Can you upload the graph image or describe the behavior near x = −7 and x = 7 (what the curve does just left and just right of those x-values)?

If you want to decide it yourself, here are two quick ways:

1) From the graph: look at points just to the left and just to the right of each vertical asymptote.
- If y goes up without bound, the limit is +∞.
- If y goes down without bound, the limit is −∞.
- Record the left- and right-hand limits at x = −7 and x = 7 and match them to the statements.

2) From an equation f(x) = N(x)/D(x): factor D(x). For an asymptote at x = a, check the sign of N(x) and D(x) for x values slightly less than a and slightly greater than a. The sign of f(x) determines whether it goes to +∞ or −∞ on each side (positive → +∞, negative → −∞). If a factor in D has even multiplicity the sign of D doesn’t change crossing a, if odd multiplicity it does.

If you send the graph or the function I’ll pick the correct statement.