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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

In the theory of relativity, the Lorentz contraction formula

L = L 0 1 v 2 / c 2

expresses the length L of an object as a function of its velocity v with respect to an observer, where L0 is the length of the object at rest and c is the speed of light. Find limv→cL and interpret the result. Why is a left-hand limit necessary?

To determine

To find: The limit of the function L as v approaches left-hand side of c and interpret the result; also explains why the left-hand limit is necessary.

Explanation

Given:

The Lorentz contraction formula, denoted by L is L01v2c2.

Where, L represents the length of an object as a function of its velocity v with respect to an observer; L0 is the length of the object at rest and c is the speed of light.

Limit Laws:

Suppose that c is a constant and the limits limxaf(x) and limxag(x) exist, then

Limit law 2: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 3: limxa[cf(x)]=climxaf(x)

Limit law 7: limxac=c

Limit law 9: limxaxn=an where n is a positive integer.

Limit law 11: limxaf(x)n=limxaf(x)n where n is a positive integer, if n is even, assume that limxaf(x)>0.

Evaluation:

Obtain the limit of the function L as v approaches left-hand side of c.

Compute, limxcL=limxcL01v2c2.

limxcL01v2c2=L0limxc1v2c2              [by limit law 3]=L0limxc(1v2c2) </

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