Verify if limit of ln(1+x)/x is 1, x-->0
- print Print
- list Cite
Expert Answers
calendarEducator since 2010
write12,544 answers
starTop subjects are Math, Science, and Business
We have to verify that lim x-->0 [ ln(1+x)/x] = 1.
substituting x = 0, we get the indeterminate form 0/0, therefore we can use the l'Hopital's rule and substitute the numerator and denominator with their derivatives.
=> lim x-->0 [ ((1/(1+x))/1]
substituting x = 0
=> 1/(1 + 0)
=> 1
This verifies that lim x-->0 [ ln(1+x)/x] = 1.
Related Questions
- Prove that limit of the function (a^x-1)/x=lna,x->0,using two methods.
- 1 Educator Answer
- How to evaluate the limit of (cos x - cos 3x) / x*sin x if x-->0 ?
- 1 Educator Answer
- Evaluate the limit of the function ln(1+x)/(sinx+sin3x) x-->0
- 1 Educator Answer
- Determine the limit of the function (sin5x-sin3x)/x, x-->0
- 1 Educator Answer
- Evaluate the limit of the function (2x-sin2x)/x^3; x-->0.
- 1 Educator Answer
In other words, we'll have to prove that:
lim [ln(1+x)]/x = 1
We'll re-write the function:
lim (1/x)*ln(1+x) = 1
We'll apply the power property of logarithms:
lim ln [(1+x)^(1/x)] = ln lim [(1+x)^(1/x)]
But, for x->0 lim [(1+x)^(1/x)] = e (remarcable limit)
lim ln [(1+x)^(1/x)] = ln e
We know that ln e = 1
So, for x->0, lim ln [(1+x)^(1/x)] = 1.
Student Answers