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Calculating Limits

Calculating Limits

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Limit of a defined Function


 »  A function `f(x)` at `x=a` is
    →  defined by limit: if `f(a)=0/0` and LHL = RHL

Limit of defined functions

plain and simple summary

nub

plain and simple summary

nub

dummy

If the left hand and right hand limits are equal then the limit is defined at that input value.

If the function evaluates to indeterminate value and the limit is defined, then the function is defined by the limits.

simple steps to build the foundation

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simple steps to build the foundation

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


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Starting on learning "Limit of defined functions". ;; With an example, calculation of limits for a defined, but discontinuous function is discussed. The conditions under which a function is defined are illustrated with examples.

Given function `f(x)=color(deepskyblue)(x^2-1)/color(coral)(x-1)`. what is `f(x)|_(x=1)`?limit of defined function

  • Substitute `x=1`
  • `(1-1)/(1-1)`
  • `0/0`
  • all the above

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

Given function `f(x)=color(deepskyblue)(x^2-1)/color(coral)(x-1)`. what is left-hand-limit `lim_(x->1-)f(x)`?left hand limit of defined function

  • `2`
  • `0/0`
  • `-delta`
  • `-2`

The answer is '`2`'.

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

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

  • `2`
  • `0/0`
  • `delta`
  • `-2`

The answer is '`2`'.

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

Given function `f(x)=color(deepskyblue)(x^2-1)/color(coral)(x-1)`.

 •  `f(1)=0/0` indeterminate value

 •  `lim_(x->1-)f(x) = 2`

 •  `lim_(x->1+)f(x) = 2`
limits of a defined function The function is defined at `x=1`.

When left-hand-limit and right-hand-limit are equal, then limit is defined.
Limit is referred as "limit of the function"

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Limit is defined: If `lim_(x->a-)f(x)` `= lim_(x->a+)f(x)` then the limits are commonly given as `lim_(x->a) f(x)`
Function is defined by Limit: If `f(a) !in RR` and `lim_(x->a-)f(x)` `= lim_(x->a+)f(x)`, then the function is defined by limit at `x=a`.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Find if the function `f(x)=color(deepskyblue)(3x^2-2x-8)/color(coral)(x-2)` is continuous or defined at `x=2`?

  • Both continuous and defined
  • Not defined
  • Not Continuous and Defined

The answer is 'Not Continuous and Defined'

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Progress

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Progress

Given function f of x = x squared minus 1 by x minus 1. What is f of x at x = 1?
substitute;x
Substitute x=1
minus
1 minus 1 by 1 minus 1
by;0
0 by 0
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 = x squared minus 1 by x minus 1. What is left hand limit limit x tending to 1 minus, f of x?
2
2
0;by
0 by 0
delta
minus delta
minus 2;-2
minus 2
The answer is "2". ;; limit x tending to 1 minus, f of x ;; equals 1 minus delta squared minus 1, by, 1 minus delta minus 1 ;; equals 1 minus 2 delta + delta squared minus 1, by, 1 minus delta minus 1;; equals minus delta into 2 minus delta, by minus delta;; equals 2 minus delta;; equals 2
Given function f of x = x squared minus 1 by x minus 1. What is right-hand-limit limit x tending to 1 +, f of x?
2
2
0;by
0 by 0
delta
delta
minus 2;-2
minus 2
The answer is "2" ;; limit x tending to 1 +, f of x ;; equals 1 + delta squared minus 1, by, 1 + delta minus 1 ;; equals 1 + 2 delta + delta squared minus 1, by, 1 + delta minus 1;; equals delta into 2 + delta, by delta;; equals 2 + delta;; equals 2
Given function f of x = x squared minus 1 by x minus 1. ;; f of 1 = 0 by 0, indeterminate value;; limit x tending to 1 minus, f of x = 2;; limit x tending to 1 +, f of x = 2;; The function is defined at x=1.
When left-hand-limit and right-hand-limit are equal, then limit is defined. Limit is referred as "limit of the function"
If the left hand and right hand limits are equal then the limit is defined at that input value. ;; If the function evaluates to indeterminate value and the limit is defined, then the function is defined by the limits.
Limit is defined : If limit x tending to a. minus, f of x equals limit x tending to a. +, f of x then limits are commonly given as limit x tending to a. f of x ;; Function is defined by limit: if f of a not in real numbers and limit x tending to a. minus, f of x equals limit x tending to a. +, f of x, then the function is defined by limit at x = a.
Find if the function f of x = 3 x squared minus 2 x minus 8 by x minus 2 is continuous or defined at x = 2?
both
Both continuous and defined
not defined
Not defined
not continuous
Not Continuous and Defined
The answer is 'Not Continuous and Defined'

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