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Math 115
Jeff Merminโs sections, Quiz 2, September 1
- Compute (x โ 1)^3.
Solution:
(x โ 1)^3 = ((x โ 1)(x โ 1)) (x โ 1) = (x(x โ 1) โ 1(x โ 1)) (x โ 1) =
x^2 โ x โ (x โ 1)
(x โ 1) = (x^2 โ 2 x + 1)(x โ 1) = x^2 (x โ 1) โ 2 x(x โ 1) + 1(x โ 1)
= x^3 โ 3 x^2 + 3x โ 1.
- Compute 2 xโ 2 2 y.
Solution: 2 xโ 2 2 y= 2(x 2 โ y)= 22 (x โ y) = x โ y.
- Compute the following limits (or state that they do not exist), using the method of your choice. No justification is required for correct answers.
(a) lim xโ 0
x^3 โ 1 2 x^3 + x^2 Solution:
lim xโ 0
x^3 โ 1 2 x^3 + x^2
lim xโ 0 x^3 โ 1 lim xโ 0
2 x^3 + x^2
= โโ since the dominant term of the denominator, x^2 , is always positive near x = 0, or, if you prefer, does not exist.
(b) lim xโ 1
x^3 โ 1 2 x^3 + x^2 Solution:
lim xโ 1
x^3 โ 1 2 x^3 + x^2
lim xโ 1 x^3 โ 1 lim xโ 1 2 x^3 + x^2
=
(c) lim xโโโ
x^3 โ 1 2 x^3 + x^2 Solution:
lim xโโโ
x^3 โ 1 2 x^3 + x^2
= lim xโโโ
x^3 โ 1 2 x^3 + x^2
1 x^3 1 x^3
= lim xโโโ
1 โ (^) x^13 2 + (^1) x
lim xโโโ
x^3 lim xโโโ
x = 12.
(d) lim xโโ
x^3 โ 1 2 x^3 + x^2 Solution:
lim xโโ
x^3 โ 1 2 x^3 + x^2
= lim xโโ
x^3 โ 1 2 x^3 + x^2
1 x^3 1 x^3
= lim xโโ
1 โ (^) x^13 2 + (^1) x
lim xโโ
x^3 lim xโโ
x = 12.
- Indicate whether the following statements are true or false. Where pos- sible, provide justification. (โTrueโ means โalways trueโ, โfalseโ means โsometimes falseโ.)
(a) lim xโ 2
x x + 1
x โ 1
= lim xโ 2
x x + 1
x โ 1
This is true. The limit plays nicely with sums, that is, we have lim xโa (f (x) + g(x)) = lim xโa f (x) + lim xโa g(x) whenever the two limits on the right exist, (which they do here). (b) If f is defined at a, then lim xโa exists.
This is false. For example, let a = 0 and consider the split function
f (x) =
0 x โค 0 1 x > 0
If you have good thoughts about this question, please talk to me about it during office hours.
