Chapter 1: Q. 92 (page 151)

Prove the $k = 1$ case of the first part of Theorem 1.31(b): that $\lim_{x \to \infty} e^{x} = \infty$ . (Hint: Given $M > 0$ , choose $N = \ln M$ . Then if $x > N = \ln M$ , we must have $x = \ln M + c$ for some positive number c. Use this to show that $e^{x} > M$ .)

Short Answer

Expert verified

The first part of Theorem 1.31(b) is proved.

$\lim_{x \to \infty} e^{x} = \infty$

Step by step solution

Step 1. Given Information

We are given a theorem 1.31(b),

$\lim_{x \to \infty} e^{x} = \infty$

Step 2. Proving the statement.

Proving the statement,

For $x > N$ ,

$x > \ln M x = \ln M + c e^{x} = e^{\ln M + c} e^{x} = M (e^{c}) x^{k} > M$

Since 'M' is arbitrary. This implies that $\lim_{x \to \infty} e^{x} = \infty$ .

Hence Proved.

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Prove the $k = 1$ case of the first part of Theorem 1.31(b): that $\lim_{x \to \infty} e^{x} = \infty$ . (Hint: Given $M > 0$ , choose $N = \ln M$ . Then if $x > N = \ln M$ , we must have $x = \ln M + c$ for some positive number c. Use this to show that $e^{x} > M$ .)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement.

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Prove the k=1 case of the first part of Theorem 1.31(b): that limx→∞ex=∞. (Hint: Given M>0, choose N=lnM. Then if x>N=lnM, we must have x=lnM+c for some positive number c. Use this to show that ex>M.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Theoretical and Mathematical Physics

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Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Prove the $k = 1$ case of the first part of Theorem 1.31(b): that $\lim_{x \to \infty} e^{x} = \infty$ . (Hint: Given $M > 0$ , choose $N = \ln M$ . Then if $x > N = \ln M$ , we must have $x = \ln M + c$ for some positive number c. Use this to show that $e^{x} > M$ .)