Can you solve this limits please? i need to help of you?

Question

Lim (x+ x^2 + x^3 +... + x^n - n)/x-1 when x ---> 1 (tell more!)
Lim ((x^3)*Sin(1/x))/(1-cos(x)) when x ---> 0
Lim xtan(x) - (p/2)*Sec(x) when x ---> p/2
Lim (Sin^4(x))/ (4 - Cos(1-cos(x))) when x ---> 0
Lim (1-(x^2))/sin(px) when x --- > 1

Answer 1

Use l'Hospital's Rule when dealing with 0/0 or intfy/infty: in such cases,

lim (f(x)/g(x)) = lim (f'(x)/g'(x)) if the latter exists (you may have to repeat this step using the second derivatives f''/g'').

In your first example, as x --> 1,

lim (x+ x^2 + x^3 +... + x^n - n)/(x-1)
= lim (1+ 2x + 3x^2 +... + nx^(n-1) )/1
= 1+ 2 + 3 +... + n = n(n+1)/2.

Answer 2

Lim (x+ x^2 + x^3 +... + x^n - n)/x-1 when x ---> 1 (tell more!)

Suppose n is 4 then (x^4 + x^3 + x^2 + x - 4)/(x-1)
becomes x^3 + 2x^2 + 3x + 4 which is 10 if x --> 1

So it looks like the limit is 1+2+3+4+5+...+n

Answer 3

Let's do that.

For the first one:
1 + 2x + 3x^2 + 4x^3 + ... + nx^(n-1)

As x ---> 1

1 + 2 + 3 + 4 + ... + n

Answer 4

Do your own homework!!!