Explain in your own words what is meant by the equation `lim_(x->2) f(x) = 5` Is it possible for this statement to be true and yet `f(2) = 3`? Explain

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Chapter 2, 2.2 - Problem 1 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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adayhare's profile pic

adayhare | College Teacher | (Level 1) Adjunct Educator

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In words, this equations says "the limit of the function 'f of x', as x approaches the value 2, is equal to the value 5". This means that as we choose values of x less than and greater than 2, the value of the function f(x) approaches 5 from the left and from the right. But the function could have a 'hole' at x = 2, and have the point value f(2) = 3. Please see the attached example.

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kspcr111's profile picture

kspcr111 | In Training Educator

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The equation `lim_(x->2) f(x) = 5` means that  for the values of x approaching to 2, the  values of the function f (x) are being  5. As we go the closer to the x values of  2, the closer the resulting f (x) values are supposed to get to 5.

 Yes,It is possible for the limit to equal 5 even though f (2) = 3.  The function` f (x) = [[(1 - x^2)]]` , the limit as x approaches 0 is 0, but f (0) = 1. so now we can create a relation as the  function

`f (x) = 5- 2 [[(1- (x-2)^2)]]`

and for this function, the limit as x approaches 2 is 5, but f (2) = 3

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