# How did earlier mathematicians calculate limits so accurately?

##
Especially things like how #lim_(x->0)# #(sinx^@/x^@)=sin1# but #lim_(x->0)# #(sinx^"rad"/x^"rad")=1#

Especially things like how

With the exception of using L'Hospital's Rule and Taylor Series (which both rely on Calculus) they would use the same techniques that we currently use.

Most notably is that the limit you have used as an example, which can easily be derived using basic geometry.

Example

This is a technique used by Archimedes circa 250BC where he used regular polygons inside and outside to obtain approximations for

The above figure shows a regular

Now,

# 360^o div 12 = 30^o => hat(AOB) = 30^o #

# C# is the midpoint of#AB => AB=2BC #

# :. COB =15^o#

By trigonometry we have:

# sin hat(COB)=(BC)/1 #

But#AB=2BC# and#hat(COB)=15^o => sin 15^'=1/2AB#

# :. AB = 2sin15^o #

Using

# cos 30^o = 1-2sin^2 15^o #

# :. sqrt(3)/2 = 1-2sin^2 15^o #

# :. 2sin^2 15^o = 1-sqrt(3)/2#

# :. sin^2 15^o = (1-sqrt(3)/2)/2#

# :. sin^2 15^o = (2-sqrt(3))/4#

# :. sin 15^o = 1/2sqrt(2-sqrt(3) #

Then he calculated the perimeter of the polygon using:

# P = 12AB #

# \ \ \ = 24sin 15^o #

# \ \ \ = 12 sqrt(2-sqrt(3) #

Then as above:

Circumference of circle

#gt # perimeter of polygon

# :. (2pi)(1) gt 12 sqrt(2-sqrt(3) #

# :. pi gt 6 sqrt(2-sqrt(3) # ..... [A]

Next Archimedes considered a 12-sided polygon that lies outside a circle of radius

As before,

# hat(DOE) = 30^o => hat(FOE)=15^o #

And,#DE=2FE#

By trigonometry:

# tan hat(FOE)=(FE)/1 #

But#DE=2FE# and#hat(FOE)=15^o => tan15^o=1/2 \ DE #

Using

# tan 30^o = (2tan15^o)/(1-tan^2 15^o) #

# :. 1/sqrt(3) = (2t)/(1-t^2) # , where#t=tan 15^o #

# :. 2sqrt(3)t = 1-t^2 #

# :. t^2+2sqrt(3)t-1 = 0 #

This is quadratic in

# t = (-2sqrt(3) +- sqrt(2sqrt()^2-4(1)(-1)) ) /2 #

# \ \ = -sqrt(3) +- 2 #

But as#t=tan 15^o > 0 => tan 15^o = 2-sqrt(3) #

And so the perimeter of this polygon is:

# P = 12 DE #

# \ \ \ = 12 tan 15^o #

# \ \ \ = 24(2-sqrt(3)) #

This time:

Circumference of circle

#lt # perimeter of polygon

# :. (2pi)(1) lt 24(2-sqrt(3)) #

# :. pi lt 12(2-sqrt(3)) # ..... [B}

Combining the results [A] and [B} Archimedes then showed that:

# 6 sqrt(2-sqrt(3)) lt pi lt 12(2-sqrt(3)) #

# :. 3.10582 ... lt pi lt 3.21539 ... #

# :. 3.106 lt pi lt 3.215 #

And of course we know that:

# 3.141592653589793238462 ... #

Archimedes used this same technique on a

As we take larger and larger

By signing up, you agree to our Terms of Service and Privacy Policy

Earlier mathematicians calculated limits using various methods, such as algebraic manipulation, geometric reasoning, and numerical approximations. They relied on techniques like the method of exhaustion, which involved finding upper and lower bounds for a function to determine its limit. Archimedes, for example, used this method to calculate the value of π. Additionally, mathematicians like Isaac Newton and Gottfried Leibniz developed calculus, which provided a systematic framework for calculating limits using differentiation and integration. These methods allowed mathematicians to accurately calculate limits and make significant advancements in the field of mathematics.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you use the epsilon delta definition to prove that the limit of #2x-4=6# as #x->1#?
- How do you find the limit of #(x/(x+1))^x# as x approaches infinity?
- How do you find the limit of #(sqrt(1+2x))-(sqrt(1-4x)] / x# as x approaches 0?
- How do you find #lim (3x)/(4x-10)# as #x->-oo#?
- How do you evaluate the limit #3x^2+4x-5# as x approaches #5#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7