How do you use a graph to determine limits?

Question

Charles Anctil · Accepted Answer

The limit of a function #f(x)# at a given point #x=a# is, essentially, the value one would expect the function #f(x)# to take on at #x=a# if one were going solely by the graph. For example, if given a graph which resembles the function #f(x) = x-1#, one might expect the function to take on the value #f(x) = 0# at #x=1#. However, the function #f(x) = (x-1)^2 /(x-1)# would also be graphed like #f(x) = x-1#, but would be undefined at #x=1#.  In the case listed above, one would analyze the situation by examining the function's behavior in the graph for #x#-values slightly above and slightly below the desired point. For this case, suppose one examines the graph at the points #x= 0, x = 0.5, x = 0.75, x = 1.25, x=1.5, x=2#. Doing this, we determine that as #x->1# from both the right and the left, #f(x) -> 0#. Thus, the two-sided limit of the function #f(x) = (x-1)^2 /(x-1)# at #x=1# is 0, though #f(1)# itself is undefined (as it takes on the form #0/0#)

Adam Adkins · Answer

undefined To use a graph to determine limits, you can follow these steps:

1. Examine the graph around the point where you want to find the limit.
2. Observe the behavior of the function as it approaches the point from both the left and right sides.
3. If the function approaches a specific value as x approaches the point from both sides, then that value is the limit.
4. If the function approaches different values from the left and right sides, or if it approaches infinity or negative infinity, then the limit does not exist at that point.
5. You can also use the graph to estimate the limit by looking at the y-values of the function as x gets closer and closer to the desired point.

Remember that a graph can provide visual insights into the behavior of a function, but it is important to verify your findings algebraically if possible.