ONE- Sided Limit PDF

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ONE – SIDED LIMITS In a graph of a given function, when a function has a break point, then the limit does not exist at the break point. For such functions, it is proper to introduce one-sided limits.

a. Right hand limit (one-sided limit from the right) is written as:

lim f ( x ) x→ a+

b. Left-hand limit (one sided limit from the left) is written as:

lim f ( x ) x →a−

c. Two-sided limit (undirected limit) is written as:

lim f ( x ) x →a

A two sided limit exists if and only if

lim f  x   lim f  x 

x  a

x a

=L

1. Find the indicated limit if it exists and sketch the graph of

-4 if x > 1   f ( x) = -1 if x = 1  2 if x < 1    Solution: a. Evaluate the one-sided limit

lim f(x) = lim- 2 x 1

x 1

=2

lim f ( x)  lim   4  x 1

x 1

= -4

limf (x ) does not exist x 1

since lim f (x )  lim f ( x ) x 1

x 1

b. The graph

y

3

x

-3

2. Find the indicated limit if it exists and sketch the graph of

6+x if x < -2  g( x ) =  2  if x -2  x Solution: a. Evaluate the one-sided limit

lim g( x)

x  -2 -

 lim -(6  x) x  -2

= 6–2 = 4

lim g ( x ) 

x  -2

lim x 2

x  -2

= (-2)2 = 4

lim g( x)  lim- g( x)

x  2

∴

x 2

lim g ( x )=4 x →2

, exists.

b. Graph g(x)

4

-6

-2

4

x...