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Thought-Process to Discover Knowledge

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summary of this topic

Limit of Trigonometric, Logarithmic, Exponential Functions

Limit of Trigonometric, Logarithmic, Exponential Functions

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Limit of Trigonometric Functions



chord length equals arc length for tiny angles

 »  `lim_(x->0) (sin x )/x = 1`

 »  `lim_(x->0) (arcsin x )/x = 1`


chord distance equals 0 compared to arch length for tiny angles

 »  `lim_(x->0) (1-cos x )/x = 0`

Limit of Trigonometric Functions

plain and simple summary

nub

plain and simple summary

nub

dummy

For small values of `x`
`sin x = x`

For small values of `x`, `x` is far greater than `1-cos x`.
`x >> (1-cos x) ~= 0`

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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Standard limits involving trigonometric function are explained.


Keep tapping on the content to continue learning.
Starting on learning "limit of trigonometric functions". ;; Standard limits involving trigonometric function are explained.

How to find the expected value for `f(x) = (sin x)/x` at `x=0`?

When a function evaluates to `0/0` at an input value, the common factors of the numerator and denominators are canceled to calculate the limit of the function at the input value. That works only if the numerator and denominator are polynomials.

When one of the numerator or denominator is a trigonometric function, how to compute the limits?

There are multiple proofs for `lim_(x->0) (sin x)/x = 1`.

 •  Substitute series expansion
`sin x = x - x^3/(3!) + x^5/(5!) + cdots`

 •  Geometrically prove that
`cos x <(sin x)/x < 1`

 •  Use the L'hospital's rule to differentiate numerator and denominator

In this, an intuitive understanding (not a proof) is given.

Consider the unit circle with angle `x` radians. intuitive understanding of sin x / x limit length of line segment `qp = sin x`
length of arc `rp` = `x`
What is the ratio `(sin x)/x =`?

  • `(text(length)(qp)) /(text(length)(rp))`
  • `(text(length)(rp)) /(text(length)(qp))`

The answer is '`(text(length)(qp)) /(text(length)(rp))`'

As `x->0`, the figure is zoomed in to the part `qp` and `rp`.intuitive understanding of sin x / x limit x tending to 0 As `x` is getting closer to `0`, what would be the lengths of line `qp` and arc `rp`?

  • Length of arc `rp` is equal to length of line `qp`
  • Length of arc `rp` is smaller than length of line `qp`

The answer is 'Length of arc `rp` is equal to length of line `qp`'

There are multiple proofs for `lim_(x->0) (1-cos x)/x = 0`.

 •  Substitute series expansion
`cos x = 1-x^2/(2!) + x^4/(4!) + cdots`

 •  Use the equality
`1-cos x = 2 sin^2 (x/2)`
and use the previous result for `(sin x)/x`

 •  Use the L'hospital's rule to differentiate numerator and denominator

In this, an intuitive understanding (not proof) is given.

Consider the unit circle with angle `x` radians. intuitive understanding of 1- cos x / x limit length of line segment `qr` = `1 - cos x`
length of arc `rp` = `x`
What is the ratio `(1-cos x)/x =`?

  • `(text(length)(qr)) /(text(length)(rp))`
  • `(text(length)(rp)) /(text(length)(qr))`

The answer is '`(text(length)(qr)) /(text(length)(rp))`'

As `x->0`, the figure is zoomed in to the part `qr` and `rp`.intuitive understanding of 1- cos x / x limit x tending to 0 As `x` is getting closer to `0`, what would be the lengths of line `qr` and `rp`.

  • Length of `qr` becomes `0` faster than length of arc `rp`
  • Length of `qr` equals the length of arc `rp`

The answer is 'Length of `qr` becomes `0` faster than length of arc `rp`'

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Limit of sin(x)/x:
`lim_(x->0) (sin x)/x = 1`

Limit of (1-cos(x))/x:
`lim_(x->0) (1 - cos x)/x = 0`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

What is `lim_(x->0) (tan x)/x`?

  • `0`
  • `1`

The answer is '`1`'
`lim_(x->0) (tan x)/x`
`quad quad = lim_(x->0) (sin x)/x xx 1/cos x`
`quad quad = 1 xx 1`

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Progress

How to find the expected value for f of x = sine x by x at x = 0? ;; When a function evaluates to 0 by 0 at an input value, the common factors of the numerator and denominators are canceled to calculate the limit of the function at the input value. That works only if the numerator and denominator are polynomials. ;; When one of the numerator or denominator is a trigonometric function, how to compute the limits?
There are multiple proofs for limit x tending to 0, sine x by x, = 1. ;; substitute series expansion, sine x = x minus x cube by 3 factorial plus x power 5 by 5 facorial et cetera ;; Geometrically prove that cos x is less than sine x by x which is less than 1 for all values of x. ;; Use lopitals rule to differentiate numerator and denominator ;; In this, an intuitive understanding is given. It is not a rigorous proof.
Consider the unit cirlce with angle x radians. ;; Leng of line segment q p = sine x ;; length of arc r p = x ;; what is the ratio sine x by x?
1
length q p by length r p
2
length r p by length q p
The answer is "length q p by length r p "
As x tending to 0, the figure is zoomed into the part q p and r p. ;; As x is getting closer to 0, what would be the lengths of line q p and arc r p?
1
Length of arc r p is equal to length of line q p
2
Length of arc r p is smaller than length of line q p
The answer is "Length of arc r p is equal to length of line q p".
For small values of x, sine x equals x
limit of sine x by x : limit x tending to 0, sine x by x = 1
There are multiple proofs for limit x tending to 0, 1 minus cos x by x = 0 ;; substitute series expansion : cos x = 1 minus x squared by 2 factorial + x power 4 by 4 factorial et cetera ;; Use the equality : 1 minus cos x = 2 sine squared x by 2 and use the previous result for sine x by x ;; use the lopitals rule to differentiate numerator and denominator. ;; In this, an intuitive understanding is given. This is not a rigorous proof.
Consider the unit circle with angle x radians. Length of the line segment q r = 1 minus cos x ;; length of arch r p = x ;;What is the ratio 1 minus cos x by x ?
1
length of q r by length of r p
2
length of r p by length of q r
The answer is "length of q r by length of r p"
As x tending to 0, the figure is zoomed into the part q r and r p. As x is getting closer to 0, what would be the lengths of line q r and r p?
1
Length of q r becomes 0 faster than length of arc r p
Length of q r equals the length of arc r p
The answer is "Length of q r becomes 0 faster than length of arc r p"
For small values of x, x is far greater than 1 minus cos x. ;; x far greater than 1 minus cos x which is approximately equals 0.
limit of 1 minus cos x by x : limit x tending to 0, 1 minus cos x by x = 0.
What is limit x tending to 0, tan x by x?
0
0
1
1
The answer is "1";; limit x tending to 0, tan x by x ;; equals limit x tending to 0, sine x by x multiplied 1 by cos x ;; equals 1 multiplied 1.

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