Chapter 4, Problem 80RE

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Using the Definition of Limits at Infinity Consider lim x → ∞ 2 x x 2 + 5 (a) Use the definition of limits at infinity to find the value of M that corresponds to ε = 0.5 .(h) Use the definition of limits at infinity to find the value of M that corresponds to ε = 0.1 .

(a)

To determine

To calculate: The value of M that corresponds to a value of 0.5 for ε as per the definition of infinite limits where limx2xx2+5.

Explanation

Given:

The expression limxâ†’âˆž2xx2+5.

Formula Used:

The statement limxâ†’âˆžf(x)=a implies that for every Îµ>0 there would exist an M>0 such that the following conditions holds whenever x>M:

|f(x)âˆ’a|<Îµ

Calculation:

Consider the provided expression:

limxâ†’âˆž2xx2+5

The limit can be found by dividing the numerator and denominator by x:

limxâ†’âˆžlimxâ†’âˆž2xx2+5=limxâ†’âˆž21+5x2=21+limxâ†’âˆž5x2=21+0=2

(b)

To determine

To calculate: The value of M that corresponds to a value of 0.1 for ε as per the definition of infinite limits, where limx2xx2+5

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