Flashcards in 2.1.6 One-Sided Limits Deck (11):
• 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.
- 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.
- 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.
Evaluate lim x→2− g(x).
Evaluate lim x→1− f(x).
Evaluate lim x→2− f(x).
p(t)= t+2, t2
Evaluate lim t→2− p(t).
Evaluate lim x→1+f(x).
Evaluate lim x→3+ h(x).
The limit does not exist.