Limits

Question 1

Name all limit laws

Answer 1

1. Sum law
2. Difference law
3. Constant multiple law
4. Product law
5. Quotient law
6. Power law
7. Root law
8. Constant law
9. Direct substitution law

Question 2

Sum law

Answer 2

lim_x-a (f(x) + g(x)) = lim_x-a f(x) + lim_x-a g(x)

Question 3

Difference law

Answer 3

lim_x-a (f(x) - g(x)) = lim_x-a f(x) - lim_x-a g(x)

Question 4

Constant multiple law

Answer 4

lim_x-a (cf(x)) = c lim_x-a f(x)

Question 5

Quotient law

Answer 5

lim_x-a (f(x)/g(x)) = (lim_x-a f(x))/(lim_x-a g(x))

Question 6

Product law

Answer 6

lim_x-a (f(x) * g(x)) = lim_x-a f(x) * lim_x-a g(x)

Question 7

Power law

Answer 7

lim_x-a (f(x)^n) = (lim_x-a f(x))^n

Question 8

Root law

Answer 8

lim_x-a sqrt(f(x)) = sqrt(lim_x-a f(x))

Question 9

Constant law

Answer 9

lim_x-a c = c

Question 10

Direct substitution law

Answer 10

If f(x) is elementary (continuous everywhere, but mainly at a), not piecewise, and f(a) is defined:

lim_x-a f(x) = f(a)

Question 11

Theorems for limits at infinity

Answer 11

e^inf = inf
e^-inf = 0
1/inf = 0
inf^p = inf
inf^-p = {-inf, p is odd} {inf, p is even}

Question 12

Indeterminate values for infinity

Answer 12

inf - inf
0 * inf
inf/inf
0^inf
1^inf
inf^0
0^0

Question 13

Infinite arithmetic

Answer 13

inf + inf = inf
-inf - inf = -inf
c + inf = inf
c - inf = -inf
c(inf) = inf ; c>0
c(inf) = -inf; c<0
(inf)(inf) = inf
(-inf)(-inf) = inf
(-inf)(inf) = -inf
sqrt(inf) = inf
1/inf = 0
c/inf = 0
c^inf = inf; c>1
c^inf = 0 ; 0<c<1
c^-inf = 0; c>0

Question 14

Solving limit with finite value

Answer 14

1. If function is absolute value, convert to a piecewise limit and solve each one sided limit
2. Check if direct substitution works
3. Check if hole or asymptote
   If hole: Solve algebraically or use l’hospital’s rule
   If asymptote: Limit does not exist, but find if it’s +- inf

Question 15

Finding +- inf for limit

Answer 15

Plug in values close to a on either side of a to find +- for each side

Question 16

Solving algebraically

Answer 16

*Only applies if limit is a hole
*Goal is to stop bottom from equaling zero
*Factor and then simplify, then use direct substitution
*If you see a square root, use conjugate pairs to move radical to denominator, then simplify
*With complex fractions, combine and simplify
*Use limit laws to simplify if needed
*As a least resort use squeeze method if you are dealing with sin and cos
*You can’t assume # * DNE = DNE, but you can assume # +DNE = DNE

Question 17

Squeeze method

Answer 17

If f(x) =< g(x) =< h(x) near x=c,
and lim_x-c f(x) = L = lim_x-c h(x)
then lim_x-c g(x) = L

Question 18

A function that’s continuous at a must:

Answer 18

1. Have f(a) be defined
2. lim_x-a f(x) exists
3. Lim_x-a f(x) = f(a)

Question 19

Things needed for squeeze theorem for sin and cos

Answer 19

-1 =< sin x =< 1
-1 =< cos x =< 1

It does not matter what is inside the function. It could be 2x or 1/x or whatever.

Question 20

Solving limits at infinity

Answer 20

1. If function is absolute value, convert to a piecewise limit and solve each one sided limit
2. Check if direct substitution works
3. a. Divide everything by largest power of x then try again
   b. Use dominance theory
   c. Use l’hospital’s rule
   d. Transform the function

Question 21

Limits of e

Answer 21

lim_x-a e^x = e^(lim_x-a x)

Question 22

Limits of logs

Answer 22

lim_x-a ln(x) = ln(lim_x-a x)

Question 23

Function theorem limits

Answer 23

If f(g(x)) is continuous at x, then lim_x-a f(g(x)) = f(lim_x-a g(x))