260 CHAPTER 14. VECTOR VALUED FUNCTIONS OF ONE VARIABLE

Therefore, from (14.6),

ddt

(|r (t)−P 1|)+ddt

(|r (t)−P 0|) = 0

showing that |r (t)−P 1|+ |r (t)−P 0|=C for some constant C.This implies the curve ofintersection of the plane with the room is an ellipse having P 0 and P 1 as the foci.

14.2.3 Leibniz’s NotationLeibniz’s notation also generalizes routinely. For example, dy

dt = y′ (t) with other similarnotations holding.

14.3 Exercises1. Find the following limits if possible

(a) limx→0+

(|x|x ,sinx/x,cosx

)(b) limx→0+

(x|x| ,secx,ex

)(c) limx→4

(x2−16x+4 ,x+7, tan4x

5x

)(d) limx→∞

(x

1+x2 ,x2

1+x2 ,sinx2

x

)2. Find

limx→2

(x2−4x+2

,x2 +2x−1,x2−4x−2

).

3. Prove from the definition that limx→a ( 3√

x,x+1) = ( 3√

a,a+1) for all a ∈ R. Hint:You might want to use the formula for the difference of two cubes,

a3−b3 = (a−b)(a2 +ab+b2) .

4. Letr (t) =

(4+ t2,

√t2 +1t3, t3

)describe the position of an object in R3 as a function of t where t is measured inseconds and r (t) is measured in meters. Is the velocity of this object ever equal tozero? If so, find the value of t at which this occurs and the point in R3 at which thevelocity is zero.

5. Let r (t) =(sin2t, t2,2t +1

)for t ∈ [0,4]. Find a tangent line to the curve parame-

terized by r at the point r (2).

6. Let r (t)=(t,sin t2, t +1

)for t ∈ [0,5]. Find a tangent line to the curve parameterized

by r at the point r (2).

7. Let r (t) =(sin t, t2,cos

(t2))

for t ∈ [0,5]. Find a tangent line to the curve parame-terized by r at the point r (2).