AP Calculus AB and BC

Limit of cot x as x Approaches 0 from the Right

The limit of cot x as x approaches 0 from the right is infinity. Cotangent is cosine over sine: the numerator tends to 1 while the denominator tends to 0 through positive values, so the quotient grows without bound. From the left the limit is minus infinity.

lim⁡x→0+cot⁡x=∞\lim_{x \to 0^+} \cot x = \infty

Settled by one-sided analysis of cosine over sine.

Reading the two factors separately

Write cotangent as a quotient and look at what each part does as xx comes down to 00 from the positive side.

cot⁡x=cos⁡xsin⁡x,cos⁡x→1,sin⁡x→0+\cot x = \frac{\cos x}{\sin x}, \qquad \cos x \to 1, \quad \sin x \to 0^{+}

A quantity approaching 11 divided by a positive quantity approaching 00 grows without bound, so the limit is +∞+\infty.

lim⁡x→0+cot⁡x=∞\lim_{x \to 0^{+}}\cot x = \infty

The side is doing real work

Approaching from the left, sine is NEGATIVE, so the same argument gives minus infinity. The two-sided limit therefore does not exist, and stating the side is not a formality here.

Why this is a vertical asymptote

A vertical asymptote at x=ax = a needs only ONE side to be unbounded. Cotangent has one at x=0x = 0, and in fact at every integer multiple of π\pi, because sine vanishes at each of them while cosine does not.

Contrast this with sin⁡xx\frac{\sin x}{x}, where the denominator also vanishes but the numerator vanishes with it, producing a removable discontinuity rather than an asymptote. Whether the numerator also goes to zero is what separates the two cases.

The mistakes students make

  • Reporting ∞\infty as the two-sided limit. The left side gives −∞-\infty, so the two-sided limit does not exist.
  • Confusing cot⁡x\cot x with arctan⁡x\arctan x. Cotangent is 1tan⁡x\frac{1}{\tan x}; arctangent is the inverse function and is bounded.
  • Writing that the limit equals infinity and stopping. Infinity is not a number, and the full answer says the limit does not exist because the function increases without bound.

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

What is the limit of cot x as x approaches 0 from the right?

It is ∞\infty: cosine tends to 11 while sine tends to 00 through positive values.

What is the limit from the left?

−∞-\infty, because sine is negative just below 00. The two-sided limit therefore does not exist.

Where else does cot x have vertical asymptotes?

At every integer multiple of π\pi, which is exactly where sine is zero and cosine is not.