lim(x->1)( x^(n+1)-(n+1)+...n)/(x-1)^2
To find the limit of the given expression as x approaches 1, we can use L'Hospital's rule, which states that if we have an indeterminate form of the type 0/0 or ∞/∞, and if the limit of the ratio of the derivatives of the numerator and denominator exists, then the limit of the original function also exists and can be found by evaluating the limit of the derivatives.
Let's apply L'Hospital's rule to the given expression.
Step 1: Take the derivative of the numerator and denominator separately.
The derivative of the numerator is (n+1)x^n - n.
The derivative of the denominator is 2(x-1).
Step 2: Evaluate the limit of the ratio of the derivatives.
lim(x->1)((n+1)x^n - n)/(2(x-1))
Now, we can plug in x = 1 into the expression:
((n+1)(1)^n - n)/(2(1-1))
= (n+1 - n)/0
We have an indeterminate form of type 0/0. Therefore, we can apply L'Hospital's rule again.
Step 3: Take the derivative of the numerator and denominator separately.
The derivative of (n+1 - n) is 0 (since it's a constant).
The derivative of 0 is still 0.
Step 4: Evaluate the limit of the ratio of the derivatives.
lim(x->1)0/0
Again, we have an indeterminate form of type 0/0. So, we can apply L'Hospital's rule once more.
Step 5: Take the derivative of the numerator and denominator separately.
The derivative of 0 is still 0.
Step 6: Evaluate the limit of the ratio of the derivatives.
lim(x->1)0/0
This time, we have an indeterminate form of type 0/0. So, we can't apply L'Hospital's rule again. Instead, we need to look for an alternative method to find the limit.
In this case, we can simplify the expression further by factoring out (x-1) from the numerator:
lim(x->1) [ (x^(n+1) - (n+1)x^n + nx^n) ] / (x-1)^2
= lim(x->1) [(x-1)(x^n - (n+1)x^(n-1) + nx^(n-1)) ] / (x-1)^2
= lim(x->1) (x^n - (n+1)x^(n-1) + nx^(n-1)) / (x-1)
= lim(x->1) (x^n - (n+1)x^(n-1) + nx^(n-1))
Now, we can plug in x = 1 into the expression:
1^n - (n+1)(1)^(n-1) + n(1)^(n-1)
= 1 - (n+1) + n
Simplifying further:
= -1 + n
Therefore, the limit of the given expression as x approaches 1 is -1 + n.