Limit x tends to 1 logx upon x-1
Answers
Answered by
1
Answer:
1
Step-by-step explanation:
Method 1 --- L'Hospital's Rule
As x-->1, the expression ( log x ) / ( x - 1 ) tends to the form 0 / 0. L'Hospital's rule applies.
The derivative of log x is 1/x. The derivative of x-1 is 1. Replacing numerator and denominator by the derivatives (L'Hospital's rule), we have:
Method 2 --- Taylor series
Let u = 1 - x. Then x = 1 - u and x --> 1 becomes u --> 0. So we need the limit, as u --> 0, of ln ( 1 - u ) / ( -u ) = - ln ( 1 - u ) / u.
Now recall the Taylor series
- ln ( 1 - u ) = u + u²/2 + u³/3 +...
Dividing by u, we have
- ln ( 1 - u ) / u = 1 + u/2 + u²/3 +...
Letting u --> 0, this tends to 1.
Similar questions
World Languages,
7 months ago
English,
7 months ago
Physics,
7 months ago
Physics,
1 year ago
English,
1 year ago