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Functions, Graphs, and Limits

Functions, Graphs, and Limits

At a Glance - One-Sided Limits

Becky has been planning her Florida vacay for months. The only thing left on her to-do list is to find a new bathing suit. She's cruising the web to find the perfect one. Eventually, she gives up on the sizing charts and hops in the car to check out what Target has to offer. She drives into the parking lot and starts looking around for a space. She finds the perfect parking spot and she decides on the best way to approach it. Assuming there aren't any parking dividers, the spot can be approached from the left (a pull through in this case) or the right. Either way, the same spot is approached. How does the parking lot at Target relate to calculus? Let's learn, fellow swim-wear seekers.

Limits can be approached from the left or the right. If we're only approaching from one direction, then it's a one-sided limit.

If x is approaching 1 from the left, we write a minus sign like this:

to show that x is approaching 1 from the negative side. We call this a left-hand limit, since x is coming from the left.

If x is approaching 1 from the right, we write a positive sign like this:

to show that x is approaching 1 from the positive side. We call this a right-hand limit, since x is coming from the right.

Sample Problem

Let

Find  and  .

When we graph this function, we find this:

If x starts out to the left of zero, we can completely ignore the chunck of the graph at y = 3. As x moves closer to zero from the left, y gets closer to 1.

Therefore, .

If x starts out to the right of zero and moves closer to zero, y gets closer to 3. Then .

Notice that the actual value of f(0) is irrelevant to calculating the limits. We don't care about what's actually happening at x = 0, just what's happening as we get super close to it.

When the left- and right-hand limits exist and are approaching the same value, we say the two-sided limit exists. Using symbols, if

= = b,

then

= b.

 

If the left-and right-hand limits are not approaching the same value, then we say the limit does not exist (DNE). If the two sides can't find any common ground, there really isn't any way to decide on a value for the limit. Compromise just won't work here.

Example 1

Let  

Does exist?


Example 2

Let  

 Does exist?


Exercise 1

Find the limit.


Exercise 2

Find the limit.


Exercise 3

Find the limit.


Exercise 4

Find the limit.


Exercise 5

For the function f(x) and specified value of a, find the left- and right-hand limits of f(x) as x approaches a.

Determine if   exists, and if so state its value.

a = 5,


Exercise 6

For the function f(x) and specified value of a, find the left- and right-hand limits of f(x) as x approaches a.

Determine if   exists, and if so state its value.

a = 0,

f(x) = |x|


Exercise 7

For the function f(x) and specified value of a, find the left-side and right-side limits of f(x) as x approaches a.

Determine if    exists, and if so state its value. 

a = 3,

 


Exercise 8

For the function f(x) and specified value of a, find the left-side and right-side limits of f(x) as x approaches a.

Determine if exists, and if so state its value.

a = -10,


Exercise 9

For the function f(x) and specified value of a, find the left-side and right-side limits of f(x) as x approaches a.

Determine if exists, and if so state its value.

a = 1,


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