i'm not sure whether the result is exacti solved the limit ln(x-4)/(x-5) and the result was 1 x->5 i'm not sure if it's exact

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

To find lim x --> 5 [ ln(x-4)/(x-5)], if we substitute x = 5 , we get ln 1/0 or 0/0 which is an indeterminate form. We can use l'Hopital's rule here and use the derivatives of the numerator and the denominator.

lim x--> 5 [ (1/(x - 4)) /1]

substitute x = 5

=> 1 / 1 = 1

The value of lim x --> 5 [ ln(x-4)/(x-5)] = 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-4)/(x-5) = ln(5-4)/(5-5) = ln1/0 = 0/0

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

lim ln(x-4)/(x-5) = lim d[ln(x-4)]/dx/d(x-5)/dx

lim d[ln(x-4)]/dx/d(x-5)/dx = lim 1/(x-4)/1

We'll substitute again x by 5 and we'll get:

lim 1/(x-4)/1 = 1/(5-4) = 1

The good news is that the result provided is exact, therefore, when x approaches to 5, the lim ln(x-4)/(x-5) = 1.

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