Limit Review Sheet

4. What are the three cases where a limit does not exist? Give examples of either functions or graphs thatrepresent these cases.(1) _______________________________________________:(2) _______________________________________________:(3) _______________________________________________:5. A function is said to be continuous at a point if for the x-value c of that point the following criteria hold:i. f (c) existsii.iii.limx→c −limx→cf (x) = limx→c +f (x) = f (c)f (x)Explain, in common terms, what each of these criteria mean. Then explain, in common terms, what it means tosay that a function is “continuous”.6. Refer to the graph of the function f from problem 3 on the previous page.a. For which x-values does the graph attain a removable discontinuity? Explain which criteria of the definitionof continuity is violated.b. For which x-values does the graph attain a nonremovable discontinuity? Explain which criteria of thedefinition of continuity is violated.7. Refer to the functions f , g , and h from problem 1 on the previous page.a. Evaluate: lim f (x) lim f (x) lim f (x)x→1 − x→1 + x→1b. Evaluate: g(0) lim g(x)x→0c. Evaluate: limx→4 − h(x)lim h(x)x→4 +lim h(x)x→4d. Discuss the continuity of all three functions. That is, determine any values of x for which the functions arenot continuous. If there are any discontinuities, clarify whether each is removable or nonremovable andthen explain which criteria is violated.

8. Discuss the continuity of the following functions:a. f (x) = x2 − 7x + 12x 2 − 4x + 3⎧⎪b. g(x) = ⎨⎪⎩1− x 2 x < 12 x = 1x − 1 x > 1⎧⎪c. h(x) = ⎨⎩⎪1− x 2 x < 1x − 1 x > 1d. How the functions in parts (b) and (c) be changed (without altering the number of parts to the piecewisefunctions) to make them continuous everywhere?9. Use your responses to the problems 5 through 9 to determine what is the key feature of a function at the pointwhere there is a removable discontinuity, as opposed to where there is a nonremovable discontinuity?