Unit 1 - Limits and Continuity

Question 1

Basic Rules of limits

Answer 1

1. lim x->a [cf(x)] = c lim x->a f(x)
2. lim x->a [f(x) +- g(x)] = lim x->a f(x) +- lim x->a g(x)
3. lim x->a [f(x)g(x)] = lim x->a [f(x)] lim x-> a [g(x)]
4. lim x ->a [f(x)]/[g(x)] = lim x->a f(x) / lim x->a g(x)

Question 2

limits of
1. sin(x) / x
2. 1-cos(x)/x
3. cos(x)
4. tan(x)/x
All of these are x –> 0

Answer 2

1. 1
2. 0
3. 1
4. 1

Question 3

Limits of:
1. lim x-> 0 (1

Question 4

Condition for finding a limit continuous

Answer 4

1. Should exist at f(x)
2. LHL = RHL
3. Limit tending should tend to the value at f(x)

Question 5

Squeeze Theorem

Answer 5

If f(x) <= g(x) <= h(x), their limits are in the same range too

Question 6

Vertical Asymptote

Answer 6

Has f(x) tending to infinity

Question 6

Horizontal asymptote

Answer 6

Has lim (x tending to infinity)

Question 6

lim (x tending infinity) x^2 / e^x =

Answer 6

0

Question 7

Discontinuities example

Answer 7

1. Hole
2. Jump
3. Infinite

Question 8

IVT

Answer 8

If a function f(x) is continuous on a closed interval [a, b], and f(a) and f(b) have opposite signs (i.e., one is positive and the other is negative), then the function must take on every value between f(a) and f(b) at some point within the interval.