Calculus/Limits/Exercises

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← Proofs of Some Basic Limit Rules Calculus Differentiation →
Limits/Exercises

Basic Limit Exercises[edit | edit source]

1.
2.
3.
4.

Solutions

One-Sided Limits[edit | edit source]

Evaluate the following limits or state that the limit does not exist.

5.
6.
7.
8.
The limit does not exist.
The limit does not exist.
9.
10.

Solutions

Two-Sided Limits[edit | edit source]

Evaluate the following limits or state that the limit does not exist.

11.
12.
The limit does not exist.
The limit does not exist.
13.
The limit does not exist.
The limit does not exist.
14.
15.
16.
17.
18.
19.
20.
The limit does not exist.
The limit does not exist.
21.
22.
The limit does not exist.
The limit does not exist.
23.
24.
25.
The limit does not exist.
The limit does not exist.
26.
27.
28.
29.
30.
31.
The limit does not exist.
The limit does not exist.
32.
The limit does not exist.
The limit does not exist.
33.
The limit does not exist.
The limit does not exist.

Solutions

Limits to Infinity[edit | edit source]

Evaluate the following limits or state that the limit does not exist.

34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.

Solutions

Limits of Piecewise Functions[edit | edit source]

Evaluate the following limits or state that the limit does not exist.

48. Consider the function

a.
b.
c.
The limit does not exist
The limit does not exist

49. Consider the function

a.
b.
c.
d.
e.
f.

50. Consider the function

a.
b.
c.
d.

Solutions

Intermediate Value Theorem[edit | edit source]

51. Use the intermediate value theorem to show that there exists a value for from . If you cannot use the intermediate value theorem to show this, explain why.
Notice is continuous from . Ergo, the intermediate value theorem applies. For all , there exists a so that .
Notice is continuous from . Ergo, the intermediate value theorem applies. For all , there exists a so that .
52. Use the intermediate value theorem to show that there exists an so that for from . If you cannot use the intermediate value theorem to show this, explain why.
Notice is continuous from . Ergo, the intermediate value theorem applies.

It is known the following is true: . From there, we can directly argue the following:

By the intermediate value theorem, if is continuous from , then there exists an so that for .
Notice is continuous from . Ergo, the intermediate value theorem applies.

It is known the following is true: . From there, we can directly argue the following:

By the intermediate value theorem, if is continuous from , then there exists an so that for .
53. Use the intermediate value theorem to show that there exists a value so that for from . If you cannot use the intermediate value theorem to show this, explain why.
Notice is not continuous for since is unbounded. Ergo, the intermediate value theorem cannot be used to solve this problem.
Notice is not continuous for since is unbounded. Ergo, the intermediate value theorem cannot be used to solve this problem.

Solutions

External Links[edit | edit source]


← Proofs of Some Basic Limit Rules Calculus Differentiation →
Limits/Exercises