   Chapter 2.6, Problem 20E

Chapter
Section
Textbook Problem

Find the limit or show that it does not exist. lim t → ∞ t − t t 2 t 3 / 2 + 3 t − 5

To determine

To find: The value of limtttt2t32+3t5.

Explanation

Result used:

Limit Laws

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

Limit law 1: limxa[f(x)+g(x)]=limxaf(x)+limxag(x)

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

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

Limit law 4: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 5: limxaf(x)g(x)=limxaf(x)limxag(x) if limxag(x)0

Limit law 6: limxa[f(x)]n=[limxaf(x)]n where n is a positive integer.

Limit law 7: limxac=c

Theorem used:

If r>0 is a rational number, then limx1xr=0.

Calculation:

Obtain the value of the function as t approaches infinity.

Consider f(t)ttt2t32+3t5 .

Divide both the numerator and the denominator by the highest power of t in the denominator. That is, t320.

f(t)=tttt322t32+3t5t32=tt32ttt322t32t32+3tt325t32=ttt12t32t322+3ttt125t32=1t1212+3t125t32

Take the limit of f(t) as t approaches infinity as follows

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