Limits and Continuity Free Online FlashCards

What does limₓ→ₐ f(x) = L mean?

A limit describes the value that f(x) approaches as x approaches a, regardless of whether f(a) is equal to that value, different, or undefined.

When does a two-sided limit exist?

A two-sided limit exists exactly when the left-hand and right-hand limits both exist and are equal.

What do one-sided limits describe?

The left-hand limit uses inputs less than a; the right-hand limit uses inputs greater than a.

What does the form 0/0 indicate?

0/0 is an indeterminate form, not a limit value. Further work, such as factoring, rationalizing, or combining fractions, is required.

What three conditions define continuity at x = a?

A function is continuous at a when f(a) is defined, limₓ→ₐ f(x) exists, and that limit equals f(a).

How do infinite limits differ from limits at infinity?

An infinite limit describes function values becoming unbounded near a finite input, while a limit at infinity describes behavior as the input becomes unbounded.

Why is x = 2 a vertical asymptote of 1/(x − 2)?

For f(x)=1/(x−2), the right-hand limit at 2 is +∞ and the left-hand limit is −∞, so x=2 is a vertical asymptote.

How do you find a rational function’s limit at infinity when degrees match?

For a rational function whose numerator and denominator have equal degree, the limit at ±∞ is the ratio of their leading coefficients.

What does continuity on a closed interval require?

On [a,b], continuity requires continuity throughout (a,b), a right-hand limit equal to f(a), and a left-hand limit equal to f(b).

What characterizes a removable discontinuity?

A removable discontinuity occurs when a finite two-sided limit exists but the function is undefined there or has a different assigned value.

What characterizes a jump discontinuity?

A jump discontinuity occurs when the finite left-hand and right-hand limits exist but are unequal.

What does the Intermediate Value Theorem guarantee?

If f is continuous on [a,b] and N lies between f(a) and f(b), then some c in [a,b] satisfies f(c)=N.