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Calculating Limits

Calculating Limits

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Limit of Functions with Indeterminate Value


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

Limit of Functions with indeterminate values

plain and simple summary

nub

plain and simple summary

nub

dummy

If a function does not have value in real numbers and the two limits are not equal then the function is indeterminate at that input value.

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simple steps to build the foundation

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


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

Given function `f(x)=tan x`. what is `f(x)|_(x=pi/2)`?limit of indeterminate function

  • `+oo`
  • `-oo`
  • `0`

The answer is '`+oo`'

`f(pi/2)`
`quad quad = tan(pi/2)`
`quad quad = oo`

Given function `f(x)=tan x`.what is left-hand-limit `lim_(x->pi/2-)f(x)`?left hand limit of indeterminate function

  • `+oo`
  • `-oo`
  • `0`

The answer is '`+oo`'

`lim_(x->pi/2-)f(x)`
`quad quad = f(pi/2-delta)`
`quad quad = tan(pi/2-delta)`
`quad quad = (sin(pi/2-delta))/(cos(pi/2-delta))`
`quad quad = (cos(delta))/(sin(delta))`
`quad quad = 1/0`
`quad quad = oo`

Given function `f(x)=tan x`.what is right-hand-limit `lim_(x->pi/2+)f(x)`?right hand limit of indeterminate function

  • `+oo`
  • `-oo`
  • `0`

The answer is '`-oo`'

`lim_(x->pi/2+)f(x)`
`quad quad = f(pi/2+delta)`
`quad quad = tan(pi/2+delta)`
`quad quad = (sin(pi/2+delta))/(cos(pi/2+delta))`
`quad quad = (cos(delta))/(-sin(delta))`
`quad quad = -1/0`
`quad quad = -oo`

Given function `f(x)=tan x`.

 •  `f(pi/2)=oo`

 •  `lim_(x->pi/2-)f(x) = oo`

 •  `lim_(x->pi/2+)f(x) = -oo`
not defined function limits The function is not continuous and not defined at `x=pi/2`.

If the function evaluates to `+-oo` or `0/0` and the limits are not equal, then the function is not defined at that input value.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Function not Defined: If `f(a) !in RR` and `lim_(x->a-)f(x)` `!=lim_(x->a+)f(x)` then the function is not defined at `x=a`.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Examine the function `f(x) = 1/(x-3)` at `x=3`.

  • Continuous
  • defined
  • not defined

The answer is 'Not defined'

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Progress

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Given function f of x = tan x. What is f of x at x = pi by 2?
plus
plus infinity
minus
minus infinity
0
0
The answer is "plus infinity" ;; f of pi by 2, equals tan pi by 2, equals infinity.
Given function f of x = tan x. What is left-hand-limit limit x tending to pi by 2 minus, f of x?
plus
plus infinity
minus
minus infinity
0
0
The answer is "plus infinity". ;; limit x tending to pi by 2 minus, f of x ;; equals f of pi by 2 minus delta ;; equals tan pi by 2 minus delta ;; equals sine pi by 2 minus delta divided by cos pi by 2 minus delta;; equals cos delta by sine delta ;; equals 1 by 0;; equals infinity
Given function f of x = tan x. What is right-hand-limit, limit x tending to pi by 2 +, f of x?
plus
plus infinity
minus
minus infinity
0
0
The answer is "minus infinity". ;; limit x tending to pi by 2 +, f of x ;; equals f of pi by 2 + delta ;; equals tan pi by 2 + delta ;; equals sine pi by 2 + delta divided by cos pi by 2 + delta;; equals cos delta by minus sine delta ;; equals minus 1 by 0;; equals minus infinity
Given function f of x = tan x. ;; f of pi by 2 = infinity ;; limit x tending to pi by 2 minus, f of x = infinity;; limit x tending to pi by 2 +, f of x = minus infinity ;; the function is not continuous and not defined at x = pi by 2.
If the function evaluates to plus or minus infinity or 0 by 0 and the limts are not equal, then the function is not defined at that input value.
If a function does not have value in real numbers and the two limits are not equal then the function is indeterminate at that input value.
function not defined: if f of a not in rel numbers, and limit x tending to a. minus f of x not equals limit x tending to a. plus f of x, then the function is not defined at x = a.
Examine the function f of x equals 1 by x minus 3 at x=3.
continuous
Continuous
defined
defined
not defined;not
not defined
The answer is "not defined"

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