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Thought-Process to Discover Knowledge

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figure-out, &
learn.

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User Guide

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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.

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User Guide

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User Guide

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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

### Calculating Limits

Voice

Voice

Home

Limit of Piecewise function

»  A function f(x) at x=a is
→  not continuous and not defined: if LHL != RHL

### Limit of piecewise functions

plain and simple summary

nub

plain and simple summary

nub

dummy

If the function evaluates to a real number, then the function is defined at that input value. The limit may not be defined.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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With an example, calculation of limits for a piecewise function is discussed. The conditions under which a function is defined are illustrated with examples.

Keep tapping on the content to continue learning.
Starting on learning "Limit of piecewise functions". ;; With an example, calculation of limits for a piecewise function is discussed. The conditions under which a function is defined are illustrated with examples.

Given function
f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}
What is f(x)|_(x=1)?

• 0.5
• -1
• +1

The answer is '1', as given by definition of the function.

Given function
f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}
what is left-hand-limit lim_(x->1-)f(x)?

• 0.5
• -1
• +1

The answer is '0.5'

lim_(x->1-)f(x)
quad quad = f(x)|_(x=1-delta)
quad quad = 0.5 quad quad quad quad (as 1-delta<1)

Given function
f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}
what is right-hand-limit lim_(x->1+)f(x)?

• 0.5
• -1
• +1

The answer is '1'

lim_(x->1+)f(x)
quad quad = f(x)|_(x=1+delta)
quad quad = 1 quad quad quad quad (as 1+delta>1)

Given function
f(x)={(1, quad text(for)quad x>=1),(0.5, quad text(for)quad x<1):}

•  f(1)=1

•  lim_(x->1-)f(x) = 0.5

•  lim_(x->1+)f(x) = 1
The function is not continuous at x=1. But, the function is defined by the given definition of the function.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Function is defined by value: If f(a) in RR, and lim_(x->a-)f(x) != lim_(x->a+)f(x), then function is not continuous but defined by the definition of the function.

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Given function
f(x)={(2x, quad text(for)quad x>2),(x^2, quad text(for)quad x=2),(|x|+2, quad text(for)quad x<2):}
Examine the function at x=2.

• continuous
• defined by limit
• defined by value
• All the above

The answer is 'All the above'.

Progress

Progress

Given function f of x equals, 1 for x greater than equal 1, 0 point 5 for x less than 1. What is f of x at x equal 1?
0;point;5
0 point 5
minus
minus 1
plus
plus 1
The answer is ' 1 ', as given by definition of the function.
Given function f of x equals, 1 for x greater than equal 1, 0 point 5 for x less than 1. What is left-hand-limit limit x tending to 1 minus, f of x?
0;point;5
0 point 5
minus
minus 1
plus
plus 1
The answer is "0.5". ;; limit x tending to 1 minus, f of x ;; equals f of x at x = 1 minus delta ;; equals 0 point 5, as 1-delta is less than 1.
Given function f of x equals, 1 for x greater than equal 1, 0 point 5 for x less than 1. What is right-hand-limit limit x tending to 1 +, f of x?
0;point;5
0 point 5
minus
minus 1
plus
plus 1
The answer is "1". ;; limit x tending to 1 +, f of x ;; equals f of x at x = 1 + delta ;; equals 1, as 1 plus delta is greater than 1.
Given function f of x equals, 1 for x greater than equal 1, 0 point 5 for x less than 1. ;; f of 1 = 1;; limit x tending to 1 minus, f of x = 0 point 5 ;; limit x tending to 1 +, f of x = 1 ;; The function is not continuous at x = 1. But the function is defined bt the given definition of the function.
If the function evaluates to a real number, then the function is defined at that input value. The limit may not be defined.
Function is defined by value: If f of a in real numbers, and limit x tending to a. minus, f of x, not equal to, limit x tending to a. +, f of x , then function is not continuous but defined by the definition of the function.
Given function f of x equals, 2x for x greater than 2, x squared for x equals 2, mod x plus 2 for x less than 2. examine the function at x = 2
continuous
continuous
limit
defined by limit
value
defined by value
all;above
All the above
The answer is "All the Above".

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