4.
Let’s see if I can explain this in words. This is not the same as just finding f (3). As you can see, there is no point there. To find the limit of f ( x ) as x approaches 3, put your finger over the picture at x = 3. With that open circle hidden, ask yourself, “what does it look like f(3) might be if you had to guess at it?” You would let your eyes follow the line right on up there really close to the hidden part. What y value is it getting closer and closer to? If you had the equation to work with, you could plug in values for x like 2.5, 2.8, 2.9, 2.99, and see what the y-values are “approaching.”
5.
Put your finger over the graph at x = -2, and pretend like you don’t know what it is. Look at BOTH sides of your finger. The graph on either side of your finger need to be pointing at the same y value. In this case, the two sides of the finger seem to agree.
Now that you have the idea of what the limit is graphically , we need to learn the algebraic ways of finding limits so that we don’t have to graph every single function to find out what is going on.
When there is no funny business [holes, jumps, asymptotes are funny business], we can just PLUG IT IN.
9.
Make sure you can do all the suggested problems. You might need to be reminded about more algebra stuff [order of operations, negative exponents, rational exponents, radicals, etc.]
10.
D. How to find a limit by simplifying There is a problem with “direct substitution” here!
If I say, “This function is continuous at x = 2,” this is what I’m saying:
“ At x = 2, this function does not have a hole, a jump, or a vertical asymptote. It could have holes, gaps, jumps, or asymptote elsewhere, but at x = 2, the function is hole-free, gap-free jump- free, and vertical-asymptote-free.”
“ This function is continuous everywhere , meaning there are no holes, no gaps, no jumps, no vertical asymptotes.”
Now that you know what a continuous graph looks like, let’s learn how to recognize continuity/discontinuity algebraically so we don’t have the draw the graph for everything.
22.
I. How to determine continuities/discontinuities
All polynomial functions are “continuous” (continuous everywhere). Examples:
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