Using Graphs to Find Limits
When evaluating the limit of a function from its graph, you need to distinguish between the function evaluated at the point and the limit around the point.
Functions like the one above with discontinuities, asymptotes and holes require you to have a very solid understanding of how to evaluate and interpret limits.
When evaluating limits graphically, your main goal is to determine whether the limit exists. The limit only exists when the left and right sides of the functions meet at a specific height. Whatever the function is doing at that point does not matter for the sake of limits. The function could be defined at that point, could be undefined at that point, or the point could be defined at some other height. Regardless of what is happening at that point, when you evaluate limits graphically, you only look at the neighborhood to the left and right of the function at the point.
limx→−∞f(x) limx→−1f(x) limx→0f(x) limx→1f(x) limx→3f(x) f(−1) f(2) f(1) f(3)
limx→−∞f(x)=0 limx→−1f(x)=DNE limx→0f(x)=−2 limx→1f(x)=0 limx→3f(x)=DNE(This is because only one side exists and a regular limit requires both left and right sides to agree) f(−1)=0 f(0)=−2 f(1)=2 f(3)=0
To see the Review answers, open this PDF file and look for section 14.2.