2.1 Limits

Limit of a Continuous Function

If ƒ(x) is a continuous function for all real numbers, then (lim_x→c)(ƒ(x))=ƒ(x)*(c)

Limits of Rational Functions

A. If ƒ(x) is a rational function given by ƒ(x)=p(x)/q(x), such that p(x) and q(x) have no common factors, and c is a real number such that q(c) = 0, then

I. (lim_x→c)(ƒ(x)) does not exist

II. (lim_x→c)(ƒ(x)) = ±∞ → x=c is a vertical asymptote

B. If ƒ(x) is a rational function given by ƒ(x)=p(x)/q(x), such that reducing a common factor between p(x) and q(x) results in the agreeable function k(x), then

I. (lim_x→c)(ƒ(x)) = (lim_x→c)(p(x)/q(x))= (lim_x→c)(k(x)) = k(c) → Hole at the point (c,k(c))

Limits of a Function as x Approaches Infinity

If ƒ(x) is a rational function given by (x) = p(x)/q(x), such that p(x) and q(x) are both polynomial functions, then

A. If the degree of p(x) > q(x), (lim_x→∞)(ƒ(x))=∞

B. If the degree of p(x)<q(x), (lim_x→∞)(ƒ(x))=∞ → y=0 is a horizontal asymptote

C. If the degree of p(x)=q(x), (lim_x→∞)(ƒ(x))=c, where c is the ratio of the leading coefficients → y=c is a horizontal asymptote

Special Trig Limits

L'Hospital's Rule

If results (lim_x→c)(ƒ(x)) or (lim_x→∞)(ƒ(x)) results in an indeterminate form (0/0,∞/∞,∞-∞,0⋅∞,1∞,∞0), and ƒ(x)=p(x)/q(x), ⇒

(lim_x→c)(ƒ(x))= (lim_x→c)(p(x)/q(x))=(lim_x→c)((p^′)(x)/(q^′)(x)) and (lim_x→∞)(ƒ(x))=(lim_x→∞)(p(x)/q(x))=(lim_x→∞)((p^′)(x)/(q^′)(x))

The Definition of Continuity

A function ƒ(x) is continuous at c if:

I. (lim_x→c)(ƒ(x)) exists

II. ƒ(c) exists

III. (lim_x→c)(ƒ(x))=ƒ(c)

Types of Discontinuities

Removable Discontinuities (Holes):

I. (lim_x→c)(ƒ(x))=L (the limit exists)

II. ƒ(c) is undefined

Non-Removable Discontinuities (Jumps and Asymptotes):

A. Jumps:

(lim_x→c)(ƒ(x)) = DNE because (lim_x→c-)(ƒ(x)) ≠ (lim_x→(c^+))(ƒ(x))

B. Asymptotes (Infinite DIscontinuities):

(lim_x→c)(ƒ(x))=±∞

Intermediate Value Theorem

If ƒ is a continuous function on the closed interval [a,b] and k is any number between ƒ(a) and ƒ(b), then there exists at least one value of c on [a,b] such that ƒ(C)=k . In other words, on a continuous function, if ƒ(a)<ƒ(b), any y value greater than ƒ(a) and less than ƒ(b) is guaranteed to exist on the function ƒ.

Average Rate of Change

The average rate of change, m, of a function ƒ on the interval [a,b] is given by the slope of the secant line.

m=(ƒ(b)-ƒ(a))/(b-a)

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