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Advanced Placement /AP Calculus AB: 2.1.6 One-Sided Limits

AP Calculus AB: 2.1.6 One-Sided Limits

Advanced Placement11 CardsCreated 7 days ago

One-sided limits examine a function's behavior from only one direction—left (−) or right (+). A limit exists at a point only if both the left-hand and right-hand limits exist and are equal. If they differ, the overall limit does not exist.

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One-Sided Limits

• 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.

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Key Terms

Term
Definition

One-Sided Limits

• 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 ...

Hover to peek or log in to view all

note 1

  • A limit exists when you can show that the function gets infinitesimally close to a certain point. It is important to note that the definiti...

Hover to peek or log in to view all

note 2

  • When working with one-sided limits, there is some notation that you need to know.

  • A small superscripted “+” or “–” above the...

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.

true

Hover to peek or log in to view all

g(x)=√3−x

Evaluate lim x→2− g(x).

1

f(x)=|x−1|

Evaluate lim x→1− f(x).

0

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AP Calculus AB: 2.1.6 One-Sided Limits
TermDefinition

One-Sided Limits

• 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.

note 1

  • 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.

note 2

  • 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.

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.

true

g(x)=√3−x

Evaluate lim x→2− g(x).

1

f(x)=|x−1|

Evaluate lim x→1− f(x).

0

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

1

p(t)= t+2, t2

Evaluate lim t→2− p(t).

8

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

3

h(x)=√9−x^2

Evaluate lim x→3+ h(x).

The limit does not exist.

f(x)=√x+5

Evaluate lim x→−4+ f(x).

1