- Find the following limits if possible
- lim
_{x→0+} - lim
_{x→0+} - lim
_{x→0−} - lim
_{x→4} - lim
_{x→3} - lim
_{x→−2} - lim
_{x→∞} - lim
_{x→∞}− 2

- lim
- Find lim
_{h→0}. - Find lim
_{x→4}. - Find lim
_{x→∞}. - Find lim
_{x→∞}. - Find lim
_{x→2}. - Find lim
_{x→∞}. - Prove Theorem 7.1.2 for right, left and limits as y →∞.
- Prove from the definition that lim
_{x→a}=for all a ∈ ℝ. Hint: You might want to use the formula for the difference of two cubes, - Prove Theorem 7.1.6 from the definitions of limit and continuity.
- Find lim
_{h→0} - Find lim
_{h→0} - Find lim
_{x→−3} - Find lim
_{h→0}if it exists. - Find the values of x for which lim
_{h→0}exists and find the limit. - Find lim
_{h→0}if it exists. Here x≠0. - Suppose lim
_{y→x+}f= L_{1}≠L_{2}= lim_{y→x−}f. Show lim_{y→x}fdoes not exist. Hint: Roughly, the argument goes as follows: Forsmall and y_{1}> x,is small. Also, forsmall and y_{2}< x,is small. However, if a limit existed, then fand fwould both need to be close to some number and so both L_{1}and L_{2}would need to be close to some number. However, this is impossible because they are different. - Suppose f is an increasing function defined on . Show f must be continuous at all but a countable set of points. Hint: Explain why every discontinuity of f is a jump discontinuity and
with f

> f. Now each of these intervalsat a point, x where a discontinuity happens has positive length and they are disjoint. Furthermore, they have to all fit in. How many of them can there be which have length at least 1∕n? - Let f=. Find lim
_{x→0}and lim_{y→0}. If you did it right you got −1 for one answer and 1 for the other. What does this tell you about interchanging limits? - The whole presentation of limits above is too specialized. Let D be the domain of a
function f. A point x not necessarily in D, is said to be a limit point of D if Bcontains a point of D not equal to x for every r > 0. Now define the concept of limit in the same way as above and show that the limit is well defined if it exists. That is, if x is a limit point of D and lim
_{y→x}f= L_{1}and lim_{y→x}f= L_{2}, then L_{1}= L_{2}. Is it possible to take a limit of a function at a point not a limit point of D? What would happen to the above property of the limit being well defined? Is it reasonable to define continuity at isolated points, those points which are not limit points, in terms of a limit as is often done in calculus books? - If f is an increasing function which is bounded above by a constant, M, show that
lim
_{x→∞}fexists. Give a similar theorem for decreasing functions.

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