   Chapter 13.1, Problem 11E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# When the area under f ( x ) =   x 2 + x from x = 0  to  x =   2 is approximated, the formulas for the sum of n rectangles using left-hand endpoints and right-hand endpoints areLeft-hand endpoints: S L = 14 3 − 6 n + 4 3 n 2 Right-hand endpoints: S R = 14 n 2 + 18 n + 4 3 n 2 Use these formulas to answer Problems 9-13. 11.   f i n d lim n → ∞ S L lim n → ∞ S R .

To determine

To calculate: The value of limnSL and limnSR where SL=1436n+43n2 and SR=14n2+18n+43n2.

Explanation

Given Information:

The provided formulae are SL=1436n+43n2 and SR=14n2+18n+43n2.

Formula used:

For functions f(x) and g(x),

limn[f(x)±g(x)]=limnf(x)±limng(x).

The value of the limit limn1n=0.

Calculation:

Consider the formulae,

SL=1436n+43n2 and SR=14n2+18n+43n2

Recall that for functions f(x) and g(x),

limn[f(x)±g(x)]=limnf(x)±limng(x).

First, calculate limnSL.

Substitute SL=1436n+43n2.

limnSL=limn(1436n+43n2)=limn143limn6n+limn43n2=1436limn1n+43limn(1n)2

Use limn1n=0 to simplify further.

limnSL=1436limn1n+43limn(1n)2=1436(0)+43(0)=1430+0=143

Thus, the value of limnSL=143

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