AP Calculus AB and BC

Limit of tan x as x Approaches pi/2 from the Left

Q: Is the limit of tan x at pi/2 infinity or does it not exist?

From the left it is +\infty and from the right it is -\infty, so the two-sided limit does not exist. Both facts are correct; they answer different questions.

Q: Where are the other asymptotes of the tangent function?

At every odd multiple of \pi/2, since those are exactly the zeros of cosine. The behaviour repeats with period \pi.

The limit of tan x as x approaches pi over 2 from the left is infinity. Writing tangent as sine over cosine explains it: the numerator tends to 1 while the denominator shrinks to zero through positive values, so the quotient grows without bound.

\lim_{x \to \frac{\pi}{2}^-} \tan x = \infty

Settled by one-sided analysis of sine over cosine.

Split the tangent

\tan x = \frac{\sin x}{\cos x}

Approaching $\pi/2$ from the left, $\sin x \to 1$ and $\cos x \to 0$ . A fixed nonzero numerator over a vanishing denominator is unbounded, so the only question left is the sign.

Just left of $\pi/2$ the angle is still in the first quadrant, where cosine is POSITIVE. So the quotient is positive and grows without bound: the limit is $+\infty$ .

Why the two sides disagree

From the right, $\cos x$ is negative, so the same argument gives $-\infty$ . The two one-sided limits differ, so the two-sided limit does not exist, and $x = \pi/2$ is a vertical asymptote of the tangent graph.

This is why an answer here MUST specify a side. Writing that the limit of $\tan x$ at $\pi/2$ is infinity without saying from which side is not a partial answer, it is a wrong one.

Not sure which technique a limit wants?

The Limit Method Chooser walks the decision from direct substitution through factoring, the conjugate, and L'Hopital, and says why each one applies or fails.

Frequently asked questions

Is the limit of tan x at pi/2 infinity or does it not exist?

From the left it is $+\infty$ and from the right it is $-\infty$ , so the two-sided limit does not exist. Both facts are correct; they answer different questions.

Where are the other asymptotes of the tangent function?

At every odd multiple of $\pi/2$ , since those are exactly the zeros of cosine. The behaviour repeats with period $\pi$ .