Evaluate the limit of the function (x^2-3x+2)/(x^2-4) using L'Hospital rule.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

As you have not specified what x is tending to, I take it as 2.

We have to find the value of lim x--> 2 [(x^2-3x+2)/(x^2-4)]

If we substitute x = 2, we get the indeterminate form 0/0. So we use L'Hopital's Rule and substitute the numerator and the denominator by their derivatives

=> lim x-->2 [(2x -3)/(2x)]

substitute x = 2

=> (4 - 3) / 4

=> 1/4

The required value of lim x--> 2 [(x^2-3x+2)/(x^2-4)] = (1/4)

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

You did not mentioned the accumulation point. Though, L'Hospital rule is used only in case of indetermination, so the accumulation point can only be a common solution of denominator and numerator of the fraction.

We'll calculate the roots of the equations:

x^2-3x+2 = 0

x1 = [3 + sqrt(9 - 8)]/2

x1 = (3+1)/2

x1 = 2

x2 = 1

We'll calculate the roots of denominator:

x^2 - 4 = 0

Since it is a difference of squares, we'll re-write it as a product:

(x-2)(x+2) = 0

x - 2 = 0

x = 2

x + 2 = 0

x = -2

We notice that the common root for both is x = 2.

We'll conclude that the accumulation point is x = 2, so x->2

We'll evaluate the limit:

lim (x^2-3x+2)/(x^2-4) = (4 - 6 + 2)/(4 - 4) = 0/0

We'll apply L'Hospital rule:

lim (x^2-3x+2)/(x^2-4) = lim (x^2-3x+2)'/(x^2-4)'

lim (x^2-3x+2)'/(x^2-4)' = lim (2x - 3)/2x

We'll substitute x by 2:

lim (2x - 3)/2x = (4 - 3)/4

lim (2x - 3)/2x = 1/4

The limit of the function is: lim (x^2-3x+2)/(x^2-4) = 1/4.

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