Unit 1: Calculus Limits, Continuity, and Discontinuities: Algebra, Theorems, and Asymptotes

What can you do to simplify limits where the denominator is equal to 0?

You can factor the numerator and denominator, then cancel any removable discontinuities.

What is a removable discontinuity?

A point where an otherwise continuous curve has a hole, which can be removed by filling the hole.

What is the Squeeze Theorem?

If g(x) ≤ f(x) ≤ h(x) for all x in an interval containing a, and lim g(x) = lim h(x) = L, then lim f(x) = L.

What is the limit of sin(x)/x as x approaches 0?

1

What is the limit of (cos(x)-1)/x as x approaches 0?

0

What is the limit of sin(ax)/x as x approaches 0?

a

What is the limit of sin(ax)/sin(bx) as x approaches 0?

a/b

What is a jump discontinuity?

Occurs when the curve 'breaks' at a particular place, with limits from the left and right existing but not matching.

What is an essential/infinite discontinuity?

A discontinuity where the curve has a vertical asymptote.

What are the conditions for continuity at x=c?

f(c) exists, the limit as x approaches c exists, and lim f(x) = f(c).

What does it mean for a function to be continuous on an interval?

It is continuous at every point on that interval.

How can you remove a discontinuity?

By redefining the function without that point in the domain, often by factoring out a common root.

What is a vertical asymptote?

A line that a function cannot cross because the function is undefined there.

What is a horizontal asymptote?

The end behavior of a function; it can be crossed.

What happens if the highest power of x is in the numerator of a rational expression?

The limit as x approaches infinity is infinity; there is no horizontal asymptote.

What happens if the highest power of x is in the denominator of a rational expression?

The limit as x approaches infinity is zero, and the horizontal asymptote is the line y=0.

What happens if the highest power of x is the same in the numerator and denominator?

The limit is the coefficient of the highest term in the numerator divided by the coefficient of the highest term in the denominator.

What does the Intermediate Value Theorem guarantee?

If f(x) is continuous on [a,b] and C is between f(a) and f(b), then there is at least one number in [a,b] such that f(x) = C.