2.1.6 One-Sided Limits

Question 1

One-Sided Limits

Answer 1

• It is sometimes useful to examine limits from strictly the left or right side. Such limits are one-sided limits. A left-handed limitis the value the function approaches only from the left (increasing). A right-handed limitis the value the function approaches only from the right (decreasing). • A limit exists only if the left-handed and right-handed limits both exist and are equal.

Question 2

note 1

Answer 2

• A limit exists when you can show that the function gets infinitesimally close to a certain point. It is important to note that the definition of a limit requires the function to approach the same point from both sides.
• Sometimes it is useful to consider the behavior of a function on only one side of a point. The value a function approaches on such an interval is called a one-sided limit.
• A one-sided limit can be considered to the left or to the right of the point in question. A limit to the left of the point is called a left-handed limit. A limit to the right of the point is called a right-handed limit.
• Notice that if the left-handed limit and the right-handed limit disagree then you have shown that the limit does not exist because the function approaches different values from the two sides.

Question 3

note 2

Answer 3

• When working with one-sided limits, there is some notation that you need to know.
• A small superscripted “+” or “–” above the domain value being approached indicates a one-sided limit. A “–” indicates a left-handed limit, because all the values used in the domain must be less than the given number. A “+” indicates a right-handed limit because all the values used must be greater than the given number.
• If the left-handed limit and the right-handed limit both exist and agree, then the limit of the function is equal to the two one-sided limits.

Question 4

True or false?
If the left-handed limit as x approaches c of a function f is equal to the right-handed limit as x approaches c of that function, then the limit as x approaches c of that function is equal to the left-handed and the right-handed limit.

Answer 4

true

Question 5

g(x)=√3−x

Evaluate lim x→2− g(x).

Answer 5

1

Question 6

f(x)=|x−1|

Evaluate lim x→1− f(x).

Answer 6

0

Question 7

f(x)={x−1, x<2
x+1, x>2
Evaluate lim x→2− f(x).

Answer 7

1

Question 8

p(t)= t+2, t2

Evaluate lim t→2− p(t).

Answer 8

8

Question 9

f(x)={x,x<1
x+2,x≥1
Evaluate lim x→1+f(x).

Answer 9

3

Question 10

h(x)=√9−x^2

Evaluate lim x→3+ h(x).

Answer 10

The limit does not exist.

Question 11

f(x)=√x+5

Evaluate lim x→−4+ f(x).

Answer 11

1