Find the limit of function f(x) given by f(x)=ln(x-1)/(x-2), x->2?

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the value of lim x-->2 [ ln (x - 1)/(x - 2)]

substituting x = 2, we get the indeterminate form 0/0; that permits the use of l'Hopital's Theorem and we can substitute the numerator and denominator with their derivatives

=> lim x-->2 [ (1/(x - 1))/1]

substitute x = 2

=> 1/(2 - 1)

=> 1

The required value of lim x-->2 [ ln (x - 1)/(x - 2)] = 1.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

First, we'll substitute x by the value of accumulation point.

lim ln(x-1)/(x-2) = ln(2-1)/(2-2) = ln1/0 = 0/0

We've get an indetermination, so, we'll apply L'Hospital rule:

lim ln(x-1)/(x-2) = lim d[ln(x-1)]/dx/d(x-2)/dx

lim d[ln(x-1)]/dx/d(x-2)/dx = lim 1/(x-1)/1

We'll substitute again x by 2 and we'll get:

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

The limit of the function f(x) is lim ln(x-1)/(x-2) = 1.

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