- #1

member 731016

- Homework Statement
- Please see below

- Relevant Equations
- Please see below

For this problem,

My solution is

##P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0## where n is a member of the natural numbers

Base case (n = 1): ##P(x) = a_0x^0 = a_0##

Thus ##\lim_{x \to \infty} \frac{P(x)}{e^x} = \lim_{x \to \infty} \frac{a_0}{e^x} = a_0 \lim_{x \to \infty} \frac{1}{e^x} = a_0 \times 0 = 0##

Algebra of limits and result ##\lim_{x \to \infty} \frac{1}{e^x} = 0##, base case ##\lim_{x \to \infty} \frac{P(x)}{e^x} = 0## holds for degree zero polynomial

For inductive hypothesis, ##\lim_{x \to \infty} \frac{a_n x^n + a_{n - 1}x^{n - 1} + \cdots + a_1 x + a_0}{e^x} = 0##

To prove inductive hypothesis, consider ##h = n + 1##

##\lim_{x \to \infty} \frac{a_hx^h + a_{h - 1}x^{h - 1} + \cdots + a_1x + a_0}{e^x} = \lim_{x \to \infty} \frac{a_{n + 1}x^{n + 1} + a_nx^n + \cdots + a_1x + a_0}{e^x} = \lim_{x \to \infty} \frac{a_{n + 1}x^{n + 1}}{e^x} + \lim_{x \to \infty} \frac{a_nx^n + \cdots + a_1x + a_0}{e^x} = a_{n + 1} \lim_{x \to \infty} \frac{x^{n+1}}{e^x} = a_{n + 1}\times 0 = 0## QED.

Note I assume ##\lim_{x \to \infty} \frac{x^{n + 1}}{e^x} = 0##

Is this proof please correct?

Thanks!

My solution is

##P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0## where n is a member of the natural numbers

Base case (n = 1): ##P(x) = a_0x^0 = a_0##

Thus ##\lim_{x \to \infty} \frac{P(x)}{e^x} = \lim_{x \to \infty} \frac{a_0}{e^x} = a_0 \lim_{x \to \infty} \frac{1}{e^x} = a_0 \times 0 = 0##

Algebra of limits and result ##\lim_{x \to \infty} \frac{1}{e^x} = 0##, base case ##\lim_{x \to \infty} \frac{P(x)}{e^x} = 0## holds for degree zero polynomial

For inductive hypothesis, ##\lim_{x \to \infty} \frac{a_n x^n + a_{n - 1}x^{n - 1} + \cdots + a_1 x + a_0}{e^x} = 0##

To prove inductive hypothesis, consider ##h = n + 1##

##\lim_{x \to \infty} \frac{a_hx^h + a_{h - 1}x^{h - 1} + \cdots + a_1x + a_0}{e^x} = \lim_{x \to \infty} \frac{a_{n + 1}x^{n + 1} + a_nx^n + \cdots + a_1x + a_0}{e^x} = \lim_{x \to \infty} \frac{a_{n + 1}x^{n + 1}}{e^x} + \lim_{x \to \infty} \frac{a_nx^n + \cdots + a_1x + a_0}{e^x} = a_{n + 1} \lim_{x \to \infty} \frac{x^{n+1}}{e^x} = a_{n + 1}\times 0 = 0## QED.

Note I assume ##\lim_{x \to \infty} \frac{x^{n + 1}}{e^x} = 0##

Is this proof please correct?

Thanks!