Functions, Graphs, and Limits
Topics
Introduction to Functions, Graphs, And Limits - At A Glance:
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 is 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 and moves closer to zero, 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 .
The actual value of f(0) is irrelevant to calculating the limits.
To find or or ,
we look at values of x near, but not equal to, 0.
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).
Example 1
Let f(x) = lr 2x &x < 0 x + 1 &x ≥ 0. . Does \lim_x\to 0f(x) exist? |
Example 2
Let f(x) = lr x + 3 &x ≤ 2 5 &x >2. Does \lim_x\to 2f(x) exist? |
Exercise 1
Find the limit.
Picture
- \lim_x\to -1^-f(x)
Exercise 2
Find the limit.
PICTURE
- \lim_x\to -1^ + f(x)
Exercise 3
Find the limit.
PICTURE
- \lim_x\to 0^-f(x)
Exercise 4
Find the limit.
PICTURE
- \lim_x\to 0^ + f(x)
Exercise 5
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 \lim_x\to af(x) exists, and if so state its value.
- a = 5, f(x) = lr 7x + 1&x ≠ 5 29&x > 5.
Exercise 6
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 \lim_x\to af(x) 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 \lim_x\to af(x) exists, and if so state its value.
\item a = 3, f(x) = lr 3x-3&x ≤ 3 3x + 3&x > 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 \lim_x\to af(x) exists, and if so state its value.
\item a = -10, f(x) = lr 3&x ≤ 4 0&x > 4.
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 \lim_x\to af(x) exists, and if so state its value.
\item a = 1, f(x) = lr x + 1 &x < 1 0 x = 1 4-2x &x >1.