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Chapter 11: Q. 45 (page 860)

Evaluate the limits in Exercises 42鈥�45.

limt鈫�0+(sintt,1-costt,(1+1t)t)

Short Answer

Expert verified

The evaluation of the limit limt鈫�0+(sintt,1-costt,(1+1t)t)is (1,0,e).

Step by step solution

01

Step 1. Given Information.

The function:

limt鈫�0+(sintt,1-costt,(1+1t)t)

02

Step 2. Apply the limits.

By applying the limit and solving,

limt鈫�0+(sintt,1-costt,(1+1t)t)=limt鈫�0+(sintt)i+limt鈫�0+(1-costt)j+limt鈫�0+(1+1t)tk =1i+sin0j+ek=i+ek=(1,0,e)

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Most popular questions from this chapter

For constants 伪,尾, and 纬, the graph of a vector-valued function of the form

r(t)=伪e纬t(cost)i+伪e纬t(sint)j+尾e纬tk,t鈮�0

For each of the vector-valued functions in Exercises ,find the unit tangent vector and the principal unit normal vector at the specified value of t.

rt=cos3t,sin3t,t=蟺4

Given a vector-valued function r(t) with domain 鈩�,what is the relationship between the graph of r(t) and the graph of r(kt), where k > 1 is a scalar?

What is the dot product of the vector functions r1(t)=x1(t),y1(t)andr2(t)=x2(t),y2(t)?

In Exercises 19鈥�21 sketch the graph of a vector-valued function r(t)=x(t),y(t) with the specified properties. Be sure to indicate the direction of increasing values oft.

Domain

t鈭�鈩�,r2=1,1,r0=-3,3,r-2=-5,-5limt鈫�-鈭�x(t)=鈭�,limt鈫�-鈭�y(t)=-鈭�,andlimt鈫�鈭�r(t)=0,0.
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