# Limits Peter Gibby Limits Limits are a way

Limits Peter Gibby

Limits • Limits are a way of asking, “what does my function approach as x approaches this value? ” • It is the foundation of calculus, because calculus deals with slopes at individual points. • Limits let us look at very specific points on a graph, and the immediately adjacent points.

How do we Use Limits? • Most of the time, the limit of the function just equals f(x) at the given x value. • For example: • Lim (2 x 2)= 2(42)=32 • X 4 • But other limits don’t work out like that. • For example: • Lim ((x 2 -1)/(x-1)) f(1)=0/0=undefined • X 1

When f(x) is Undefined • We look at the values coming up to that point. We could graph it, or just plug in numbers close to it. f(0. 9) f(0. 99) f(1. 01) f(1. 1) 1. 900 1. 990 undef 2. 010 2. 100 • We can see that as x comes closer to 1, the values on either side approach 2. So we say: • Lim ((x 2 -1)/(x-1))=2 • X 1

Infinity • Functions can be interpreted as going on to infinity. Unless there is a variable in the denominator to counteract a variable in the numerator, the limit will be infinity: • Lim 2 x = ∞ • X ∞ • Lim (4 x)/x= 4 • X ∞ • We can imagine this if we use algebra to divide out the x.

Continue on Infinity • Lim (2 x+920, 123)/x • X ∞ • We can distribute the x to get a new function: 2+920, 123/x. But when we bring x to ∞, the second value is zero. You can think of it being as close to 0 as possible. So when there is a limit to ∞, any constants become irrelevant. • Lim (2 x+920, 123)/x=2 • X ∞

YAY! • Looks like you have completed my “ 4. Trigonometry” section! • If you need more practice on any particular section, return to www. perfectmath. weebly. com and find the practice for that section. Otherwise wait for “ 5. Calculus” or “ 6. Probability and Statistics” to be added! • (Have some more math cake!)

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