Prove that limx→0+1x=∞, with these steps: (a) What is the M–δ statement that must be shown to prove thatlimx→0+1x=∞? (b) Argue that x ∈ (0, 0 + δ) if and only if 0 < x < δ. (c) Argue that 1x ∈ (M,∞) if and&...

Part (a) The infinite limit, limx→0+1x=∞ means that for all M > 0, there exists δ > 0 such that if x ∈ (0, 0 + δ), then 1...

Q. 70 - Calculus | StudyQuestionHub

Q. 70

Question

$Prove that \lim_{x \to 0^{+}} \frac{1}{x} = \infty, with these steps : (a) What is the M - δ statement that must be shown to prove that \lim_{x \to 0^{+}} \frac{1}{x} = \infty ? (b) Argue that x \in (0, 0 + δ) if and only if 0 < x < δ . (c) Argue that \frac{1}{x} \in (M, \infty) if and only if x < \frac{1}{M} . You may assume that M > 0 . (d) Given any particular M > 0, what value of δ would guarantee that if 0 < x < δ, then x < \frac{1}{M} ? Your answer will depend on M . (e) Put the previous four parts together to prove the limit statement .$

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Answer

$Part (a) The infinite limit, \lim_{x \to 0^{+}} \frac{1}{x} = \infty means that for all M > 0, there exists δ > 0 such that if x \in (0, 0 + δ), then \frac{1}{x} \in (M, \infty) . Part (b) x \in (0, 0 + δ) is equivalent to 0 < x < δ . Part (c) For M > 0, \frac{1}{x} \in (M, \infty) means that M < \frac{1}{x} < \infty, which means 0 < x < \frac{1}{M} . Part (d) For a given δ > 0, choose M = \frac{1}{δ} Part (e) Given any δ > 0, let M = \frac{1}{δ} If x \in (0, 0 + δ) means that 0 < x < δ, from part (b) . which is same as 0 < x < \frac{1}{M} from our choice in part (d) . From part (c), we get, M < \frac{1}{x} < \infty and thus \frac{1}{x} \in (M, \infty) .$

1Part (a) Step 1. The M - δ statement

$The infinite limit, \lim_{x \to 0^{+}} \frac{1}{x} = \infty means that for all M > 0, there exists δ > 0 such that if x \in (0, 0 + δ), then \frac{1}{x} \in (M, \infty) .$

2Part (b) Step 1. Argument regarding δ

$Now, x \in (0, 0 + δ) is equivalent to 0 < x < δ .$

3Part (c) Step 1. Argument regarding M

$For M > 0, \frac{1}{x} \in (M, \infty) means that M < \frac{1}{x} < \infty, which means 0 < x < \frac{1}{M} .$

4Part (d) Step 1. Relationship between M - δ

$For a given δ > 0, choose M = \frac{1}{δ}$

5Part (e) Step 1. Proof of the limit statement

$Given any δ > 0, let M = \frac{1}{δ} If x \in (0, 0 + δ) means that 0 < x < δ, from part (b) . which is same as 0 < x < \frac{1}{M} from our choice in part (d) . From part (c), we get, M < \frac{1}{x} < \infty and thus \frac{1}{x} \in (M, \infty) .$

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