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Material Type: Lab; Professor: Herbekian; Class: Analytic Geo & Calc; Subject: Mathematics; University: Cuesta College; Term: Unknown 1999;
Typology: Lab Reports
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math 265A introduction to limits of functions TI83/
In this lab we shall investigate the behavior of a function f near a specified point. While this is sometimes a straightforward process, it can also be quite subtle. In many cases, the process for finding a limit must be applied carefully. By gaining an intuitive feel for the notion of limits, you will be laying a solid foundation for success in calculus.
x
x f x.
a) By successive evaluation of f at x = 1.9, 1.99, 1.999, 1.9999, what do you think happens to the values of f as x increases towards 2?
b) Do a similar evaluation of f for values of x slightly greater than 2. For example, let x = 2.1, 2.01, 2.001, 2.0001.
As a shorthand and anticipating a forthcoming definition, we shall describe what you found in parts (a) and (b) by writing
2
→
f x x
lim
4
2
→ (^) x
x x
c) Note that in this particular case you could have “ CHEATED ” by immediately evaluating f at 2. Use a graphing utility to graph f. Use a window setting with XMin = -4, XMax = 4, YMin = -4, and YMax = 25.
a) Study this situation experimentally as you did in parts (a) and (b) of problem 1, but this time choosing x values close to 1.
x 1
lim →
b) What happens when you try to “ CHEAT ” as was done in part (c) of problem 1? Use a window setting with XMin = -4.7, XMax = 4.7, YMin = -1, and YMax = 11.4.
Do you see why “ CHEATING ” DOESN’T PAY in this situation?
There are situations in which direct evaluation at the specified point is possible and actually give the limit. These give rise to a concept called continuity. However, there are many important situations that arise in calculus when this technique will not work.
a)
x
x x
sin 10 lim → 0
x
x^1 0
lim 1 + →
a) 2
lim → (^2) x −
x x
b) (^) ⎟ ⎠
x → (^) x
lim sin 0