   Chapter 2.3, Problem 26E

Chapter
Section
Textbook Problem

Evaluate the limit, if it exists. lim t → 0 ( 1 t − 1 t 2 + t )

To determine

To evaluate: The limit of the function limt0(1t1t2+t).

Explanation

Direct substitution property:

If f is a polynomial or a rational function and a is in the domain of f, then limxaf(x)=f(a).

Fact 1:

If f(x)=g(x) when xa, then limxaf(x)=limxag(x), provided the limit exist.

Let f(t)=1t1t2+t (1)

Note 1:

The direct substitution method is not applicable for the function f(t) since the function f(0) is in indeterminate form when t=0. That is,

f(0)=020(02+0)=00

Note 2:

The Quotient rule is not applicable for the function f(t) as the limit of the denominator is zero. That is,

limt0(t(t2+1))=limt0tlimt0(t2+1) (by limit law 4)=limt0t(limt0(t2)+limt0(1)) (by limit law 1)=(0)(02+1) (by limit law 8,9,7)=0

Note 3:

The limit may be infinite or some finite value when both the numerator and the denominator approach to 0

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