SOLUTION: PROVING
A. limit as delta x approaches to zero cos delta x-1/delta x
B. limit as delta x approaches to zero sin delta x/delta x
Algebra.Com
Question 1131261: PROVING
A. limit as delta x approaches to zero cos delta x-1/delta x
B. limit as delta x approaches to zero sin delta x/delta x
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
.jpg)
Segment SC = )
by definition (note unit circle). Altitude of triangle OSU is and the base of triangle OSU is 1, so the area of triangle OSU is }{2})
.
The area of a circle sector is given by 
where 
is the subtended angle. Hence, the area of sector OSU is simply 
Segment TU = )
by definition of the tangent function. (TU divided by OU, but OU measures 1). Then the area of triangle OTU is }{2})
.
From the diagram, the area of triangle OSU is less than or equal to the area of sector OSU which in turn is less than or equal to triangle OTU. Hence:
}{2}\ \leq\ \frac{x}{2}\ \leq\ \frac{\tan(x)}{2})
Multiply by })
}\ \leq\ \frac{1}{\cos(x)})
Take the reciprocal and reverse the sense of the inequalities:
\ \leq\ \frac{\sin(x)}{x}\ \leq\ 1)
But \ =\ 1)
and 
Therefore, by the Squeeze Theorem:
}{x}\ =\ 1)
John
My calculator said it, I believe it, that settles it
RELATED QUESTIONS
Prove that the limit as x approaches 5 of f(x) = ax+b is 5a+b using delta-epsilon proof. (answered by rothauserc)
could anyone help me with this qustion please.
find delta y/delta x, give the following... (answered by Alan3354)
I didn't see a Calculus section so I'm posting my question here. I need to know what I'm (answered by stanbon)
Given x delta y = x+y/x-y
8 delta w = 4/3
solve for delta w
(answered by jim_thompson5910)
P(5,7) to Q(x,y) what is the delta x and delta y from P to... (answered by stanbon)
f(x)= f(x + delta(x)) - f(x)/ delta(x) how was this function... (answered by stanbon)
Find delta x and delta y for the points (-4,6) and... (answered by Alan3354)
Apply the delta method to find derivatives of the following problem:
f(x)=... (answered by Alan3354)
What are the steps to solve a(x+delta... (answered by tommyt3rd)