Unit 0: Limits and Continuity

Question 1

conditions for continuity

Answer 1

1) lim f(x) must exist
x –> a
2) f(a) must exist
3) f(a) = lim f(x)
x –> a

Question 2

limits of trig functions

Answer 2

lim sinx/x = 1
x –> 0
lim tanx/x = 1
x –> 0
lim cosx/x = 0
x –> 0

Question 3

lim [f(x) +/- g(x)]
x –> a

Answer 3

= lim f(x) +/- lim g(x)
x –> a x –> a

Question 4

lim c * f(x)
x –> a

Answer 4

= c * lim f(x)
x –> a

Question 5

lim f(x) * g(x)
x –> a

Answer 5

= lim f(x) * lim g(x)
x –> a x –> a

Question 6

lim f(x) / g(x)
x –> a

Answer 6

= lim f(x) / lim g(x)
x –> a x –> a

Question 7

lim f^n(x)
x –> a

Answer 7

= [lim f(x)]^n for n>0
x –> a

Question 8

Squeeze theorem

Answer 8

given two familiar functions, f and g, are continuous, if…
1) f(x)</=h(x)</=g(x) on an interval containing a, and…
2) lim f(x) = lim g(x) = L
x –> a x –> a
then by squeeze theorem…
lim h(x) = L
x –> a