Limits and Continuity (Chapter 4 & 5)

Question 1

Evaluate the limit of the following graph as x aproaches 5

Answer 1

10

Question 2

Evaluate the following limit

Answer 2

1

A basic limit to remember

Question 3

Evaluate the following limit

Answer 3

a

A basic limit to remember

Question 4

Utilising basic limits

Answer 4

Question 5

Answer 5

Therefore, if indeterminate, then you must find a way to solve it

Question 6

Finish the following sentence

In the following case, we say f(x) diverges to __

Answer 6

Positive infinity

Question 7

Finish the following sentence

In the following case, we say f(x) diverges to __

Answer 7

Negative infinity

Question 8

Which cases should you pay close attention to left and right hand side limits?

Answer 8

In piecewise functions

Question 9

Answer 9

Question 10

Answer 10

Question 11

Answer 11

Note sine function will always be between [-1, 1] no matter of whats in the brackets

Question 12

Answer 12

Question 13

Answer 13

Question 14

True or false?

The ‘highest power method’ only works when x aproaches infinity

Answer 14

True

Note how you can easily just solve this

Question 15

Answer 15

Question 16

Answer 16

Question 17

Evaluate the following limit

Answer 17

1

You just have to remember this one

Note that sinθ ≈ θ for small values of θ. And so it basically becomes x/x=1

Question 18

Evaluate the following limit

Answer 18

1/2

You just have to remember this one

Question 19

Evaluate the following limit

Answer 19

1

You just have to remember this one

However note the similarity between this and first principles

Question 20

When can we say a function, f is continuous at a?

Answer 20

Question 21

Explain why f has discontinuity at x=0

Answer 21

Question 22

Answer 22

f(0) is not defined

Question 23

Explain why the following function is discontinuous at x=0

Answer 23

Question 24

Finish the following sentence

We say that f is continuous on the open interval (a, b), if ___

Answer 24

f is continuous for all x ∈ (a, b)

note (a, b) is an interval not a tuple

Question 25

# Finish the following sentence If f is continuous on the closed interval [a, b], then f is continuous on (a, b) and ___

Answer 25

Question 26

# True or false? f(x)=|x| is a continuous function

Answer 26

True

Question 27

# True or false? Adding _and_ subtracting two continuous functions gives a new continuous function

Answer 27

True

Question 28

# True or false? Multiplying two continuous functions creates a new continuous function but dividing does not

Answer 28

False. They both create continuous functions

Question 29

# Give a formal definition to the following sentence: "If a function, f is continuous at x=g(x) and a function, g is continuous at x=a then f ∘ g is also continuous at x=a"

Answer 29

Question 30

Prove the following:

Answer 30

Question 31

What does the intermediate value theorum state?

Answer 31

If f is continuous on the interval [a, b] and N is any number between f(a) and f(b) where f(a)≠f(b). Then there exists c ∈ (a, b) such that f(c)=N

Question 32

Answer 32