   Chapter 13.1, Problem 3E

Chapter
Section
Textbook Problem

# Find the limit.3. lim t → 0 ( e − 3 t i  +  t 2 sin 2 t j  + cos 2 t   k )

To determine

To find: Limit of vector function e3ti+t2sin2tj+cos2tk .

Explanation

Formula used:

Write the expression for vector function r(t) .

r(t)=f(t)i+g(t)j+h(t)k (1)

Here,

f(t) , g(t) , and h(t) are component functions of r.

Write the expression for limit of vector function (limtar(t)) .

limtar(t)=(limtaf(t))i+(limtag(t))j+(limtah(t))k (2)

Compare the vector function e3ti+t2sin2tj+cos2tk with equation (1).

f(t)=e3tg(t)=t2sin2th(t)=cos2t

Substitute e3ti+t2sin2tj+cos2tk for r(t) , e3t for f(t) , t2sin2t for g(t) , cos2t for h(t) , and 0 for a in equation (2),

limt0(e3ti+t2sin2tj+cos2tk)=(limt0e3t)i+(limt

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