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summary of this topic

Calculating Limits

Calculating Limits

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Limit of a Continuous Function


 »  A function `f(x)` at `x=a` is
    →  continuous: if `f(a)` = LHL = RHL

Limit of continuous functions

plain and simple summary

nub

plain and simple summary

nub

dummy

Function is continuous if all three values are equal.

simple steps to build the foundation

trek

simple steps to build the foundation

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With an example, calculation of limits for a continuous function is discussed. The condition under which a function is continuous is illustrated with examples.


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Starting on learning "limit of continuous functions". ;; With an example, calculation of limits for a continuous function is discussed. The condition under which a function is continuous is illustrated with examples.

Given function `f(x)=1/(x^2+1)`. what is `f(x)|_(x=1)`?limit of continuous function

  • Substitute `x=1`
  • `1/(1+1)`
  • `1/2`
  • all the above

The answer is 'All the above'. The options provide the steps to evaluate `f(1)`.

Given function `f(x)=1/(x^2+1)`. what is left-hand-limit `lim_(x->1-)f(x)`?left hand limit of continuous function

  • Substitute `x=1`
  • Substitute `x=1+delta`
  • Substitute `x=1-delta`
  • all the above

The answer is 'Substitute `x=1-delta`'.

`lim_(x->1-)f(x)`
`quad quad = 1/((1-delta)^2+1)`
`quad quad = 1/(1-2delta+delta^2+1)`
`quad quad = 1/(2-2delta+delta^2)`
`quad quad = 1/2 quad quad quad quad` (substituting `delta=0`)

Given function `f(x)=1/(x^2+1)`. what is right-hand-limit `lim_(x->1+)f(x)`?right hand limit of continuous function

  • Substitute `x=1`
  • Substitute `x=1+delta`
  • Substitute `x=1-delta`
  • all the above

The answer is 'Substitute `x=1+delta`'.

`lim_(x->1+)f(x)`
`quad quad = 1/((1+delta)^2+1)`
`quad quad = 1/(1+2delta+delta^2+1)`
`quad quad = 1/(2+2delta+delta^2)`
`quad quad = 1/2 quad quad quad quad` (substituting `delta=0`)

Given function `f(x)=1/(x^2+1)`.
 •  `f(1)=1/2`
 •  `lim_(x->1-)f(x) = 1/2`
 •  `lim_(x->1+)f(x) = 1/2`
defined function limits The function is continuous at `x=1`.

Note: Though, as part of the topic, the functions that are discontinuous or indeterminate or non-differentiable are mainly handled, finding limits is not only for those kind of functions. Any function at any point can be evaluated for limits.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Function is continuous: if `f(a)` `= lim_(x->a-)f(x)` `=lim_(x->a+)f(x)`, then the function is continuous at `x=a`.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Given function `f(x) = sin x`, is it continuous at input values `x= 0, pi/2, pi`?

  • Continuous at `0` and `pi/2` only
  • Continuous at all three input values
  • Continuous at only `0`
  • not continuous at any one of the given values

The answer is 'Continuous at all three input values'

your progress details

Progress

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Progress

Given function f of x = 1 by x squared + 1. What is f of x at x =1?
substitute;x;equal
Substitute x=1
plus;1+1;+
1 by 1 + 1
2;half
1 by 2
all;above
all the above
The answer is "All the above" The options provide the steps to evaluate f of 1.
Given function f of x = 1 by x squared + 1. What is left-hand-limit limit x tending to 1 minus, f of x?
1
Substitute x=1
plus;+
Substitute x=1+delta
minus;-
Substitute x=1 minus delta
all;above
all the above
The answer is "Substitute x = 1 minus delta" ;; limit x tending to 1 minus, f of x ;; equals 1 by 1 minus delta squared +1 ;; equals 1 by 1 minus 2 delta + delta squared + 1 ;; equals 1 by 2 minus 2 delta + delta squared ;; equals 1 by 2, by substituting delta equals 0.
Given function f of x = 1 by x squared + 1. What is right hand limit limit x tending to 1 +, f of x?
1
Substitute x=1
plus;+
Substitute x=1+delta
minus;-
Substitute x=1 minus delta
all;above
all the above
The answer is "substitute x = 1 + delta" ;; limit x tending to 1 +, f of x ;; equals 1 by 1 + delta squared, +1 ;; equals 1 by 1 + 2 delta + delta squared + 1 ;; equals 1 by 2 + 2 delta + delta squared ;; equals 1 by 2, by substituting delta equals 0.
Given function f of x = 1 by x squared + 1. ;; f of 1 = 1 by 2;; limit x tending to 1 minus, f of x = 1 by 2;; limit x tending to 1 +, f of x = 1 by 2;; The function is continuous at x=1. ;; Note: Though, as part of the topic, the functions that are discontinuous or indeterminate or non-differentiable are mainly handled, finding limits is not only for those kind of functions. Any function at any point can be evaluated for limits.
Function is continuous if all three values are equal.
Function is continuous: if f of a equals, left-hand-limit, equals right-hand-limit, then the function is continuous at x=a.
Given function f of x = sine x. Is it continuous at input values x equals 0, pi by 2, pi?
only
continuous at 0 and pi by 2 only
all;three;input
Continuous at all three input values
only 0
Continuous at only 0
not;given;any;
not continuous at any one of the given values
The answer is 'Continuous at all three input values'

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