2.1.4 The Limit Laws, Part I

Question 1

note

Answer 1

• Since limits are just numbers, a lot of the properties of real numbers also apply to limits.
• Taking the limit of a function is an operation, but the resulting limit is just a number. Therefore, it makes sense that limits have a lot of the same properties that numbers do.
• The limit of a sum of two functions is equal to the sum of the limits.
• The limit of a difference of two functions is equal to the difference of the limits.
• The limit of a product of two functions is equal to the product of the limits.
• The limit of a quotient of two functions is equal to the quotient of the limits, provided that the denominator does not equal zero.
• The limit of a function multiplied by a constant is equal to the constant multiplied by the limit.
  In addition, the limit of a function raised to a power is equal to the limit raised to that power.

Question 2

Suppose you are told that lim x→1  f(x)=3 and lim x→1  g(x)=−1. What is the value of lim x→1  [f(x)+2g(x)]?

Answer 2

1

Question 3

Given lim x→cf(x)=2 and lim x→cg(x)=4, evaluate lim x→c[2f(x)−g(x)].

Answer 3

0

Question 4

Given lim x→ 4 f(x)=2 and lim x→ 4 g(x)=3,evaluate lim x→ 2 [f(x)−g(x)/2f(x)].

Answer 4

The limit cannot be determined from the information given.

Question 5

Is the following equation true for all values of x, a, and c ? limx→ a [c⋅f(x)+g(x)]=c[limx→ af(x)+g(x)]

Answer 5

no

Question 6

Given lim x→2f(x)=3 and lim x→2g(x)=2,evaluate lim x→23f(x)−g(x)/g(x).

Answer 6

7/2

Question 7

Determine the limit (if it exists):
lim x→0 sinx/2x
Hint: lim x→0 sinx/x=1

Answer 7

1/2