Limit

[canvas]

[math]\lim_{x\to0}\frac{(1+x)^n-1-nx}{x^2}=...\\\\\text{do you have any ideas guys? I'm stuck}\\\text{but looking at the numerator what comes to my mind is:}\\\\\text{Bernoulli's inequality:}\\\\(1+x)^n\geq1+nx,\,\,n\geq0, x>-1\\\\\text{could it be used here or it's just a dead end?}[/math]

 

[BigBeachBanana]

How about l'hopital?

 

[lex]

Use the Binomial expansion for (1+x)^n, then simplify the expression and you will be able to compute the limit directly.

 

[BigBeachBanana]

Use the Binomial expansion for (1+x)^n, then simplify the expression and you will be able to compute the limit directly.

That assumes n is an integer, but it's probability is.

 

[canvas]

Use the Binomial expansion for (1+x)^n, then simplify the expression and you will be able to compute the limit directly.

thank you sir, that was a perfect hint, the answer is:

[math]\lim_{x\to0}\frac{(1+x)^n-1-nx}{x^2}=\frac{1}{2}n(n-1)[/math]

 

[canvas]

How about l'hopital?

2 times rule l'hospital also works

 

[lex]

Use the Binomial expansion for (1+x)^n, then simplify the expression and you will be able to compute the limit directly.

That assumes n is an integer, but it's probability is.

No. It is true for negative and fractional indices too.