Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.

User Guide

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
think,
figure-out, &
learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn.

User Guide

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Limit of Trigonometric, Logarithmic, Exponential Functions

Voice

Voice

Home

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

Support Nubtrek

You are learning the free content, however do shake hands with a coffee to show appreciation.
To stop this message from appearing, please choose an option and make a payment.

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. 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. 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. 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. 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

Progress

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.

we are not perfect yet...