2.2 Finding Limits Numerically

Example Consider the function: What is the value of the function when x=1? Using limits, we can determine what the value of the function “should be” even for values that are undefined.

Finding a Limit To determine what the value of a function “should be” at a point, we ask ourselves, “Self, what happens to f(x) as x gets really, really, really, really, superdy-duperdy close to c, but never actually reaches it?”

Clearly the missing value is “2”.

Notation In general, we denote the limit as x approaches a specific value c as: In reference to the last problem:

Example: Consider the following function as x approaches -2: It appears that the values for f(x) are getting larger in the negative direction as x gets closer to -2… but only on the right hand side of -2.

What it Means The last result means that there is a vertical asymptote at x = -2. How do we know if the value(s) being discluded from the domain represent a V.A. or a hole in the graph? Any value(s) for x that cause both the numerator and the denominator to be zero will result in a hole in the graph. Any value(s) for x that cause just the denominator to be zero will result in a V.A.

The difference

Kickin’ It Up A Notch Check this one out: What is the value of f(x) when x=0? What is the limit as x approaches 0?

Homework: Pg. 74 #1-10 [3], 21, 25