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MATH 1200 8 Dec 2016

LIMITS

Tangent Lines

A tangent line is a line that locally touches a function at only one point. To ﬁnd the slope of a tangent line, at a

given point a:

m= lim

h→0

f(a+h)−f(a)

h

Anormal line is the line perpendicular to the tangent; they have a slope of −1/m.

Limit Rules

Assuming that limx→af(x)and limx→ag(x)both exist, where a, c ∈R.

1. limx→a[f(x)±g(x)] = limx→af(x)±limx→ag(x)

2. limx→acf(x) = c·limx→af(x)

3. limx→af(x)g(x) = limx→af(x)·limx→ag(x)

4. limx→a

f(x)

g(x)=limx→af(x)

limx→ag(x)

5. limx→ah(f(x)) = h(limx→af(x)), for h(x)continuous.

Factor Theorem

If f(p) = 0 for any polynomial f(x),(x−p)is a factor of f(x).

Ex. Find limx→2

2x4+ 17x3+ 31x2−62x−168

6x5+ 5x4−153x3+ 8x2+ 660x−400.

If you directly substitute x= 2 into the above function, you get 0

0. This means that (x−2) is a factor for

both polynomials in the function. Factor (x−2) out of each polynomial, using synthetic division, then try direct

substitution again.

One-sided Limits

Functions can sometimes be discontinuous at certain points, this is especially true for piece-wise functions. A limit

exists only when a function is continuous, this is determined by evaluating a limit approaching xfrom both the left

and right.

limx→a−f(x) = L−means the limit of f(x)approaching xfrom the left.

limx→a+f(x) = L+means the limit of f(x)approaching xfrom the right.

If L−̸=L+, the limit doesn’t exist.

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