Prove the following limits when c ∈ N (analysis help):

[cooljosh2k2]

Homework Statement

Prove that when c ∈ N:

a)
lim [\sqrt[c]{1+x} - 1]/x = 1/c
x->0

b)
lim [(1+x)^r - 1]/x = r
x->0

,where r = c/n

The Attempt at a Solution

My approach for a and b are pretty similar, i get stuck at the same point.

For a, i multiplied the numerator and the denominator by [(1+x)^(1/c) + 1] to get rid of the radical on top. This left me with:

Lim 1/[(1+x)^(1/c)+1]
x->0

This is where I am stuck, if i find the limit of the numerator and divide it by the limit of the denominator, i don't get 1/c. So I am clearly doing something wrong.

As for part c, i pretty much use the same approach and get:

Lim 1/[(1+x)^(r)+1]
x->0

which also doesn't lead to the answer.

Please help.

 

[Mark44]

Homework Statement

Prove that when c ∈ N:

a)
lim [\sqrt[c]{1+x} - 1]/x = 1/c
x->0

b)
lim [(1+x)^r - 1]/x = r
x->0

,where r = c/n

The Attempt at a Solution

My approach for a and b are pretty similar, i get stuck at the same point.

For a, i multiplied the numerator and the denominator by [(1+x)^(1/c) + 1] to get rid of the radical on top.

This works for square roots, but not for cube roots, fourth roots, etc.

(a - b)(a + b) = a² - b²
If a is the square root of something, a² will have the radical eliminated.

(a - b)(a² + ab + b² = a³ - b³
This time, if a is the cube root of something, a³ will have the radical eliminated.

(a - b)(a³ +a²b + ab² + b³) = a⁴ - b⁴
In this example, if a is the fourth root of something, a⁴ will have the radical eliminated.

Notice that there is a pattern here.

This left me with:

Lim 1/[(1+x)^(1/c)+1]
x->0

This is where I am stuck, if i find the limit of the numerator and divide it by the limit of the denominator, i don't get 1/c. So I am clearly doing something wrong.

As for part c, i pretty much use the same approach and get:

Lim 1/[(1+x)^(r)+1]
x->0

which also doesn't lead to the answer.

Please help.

 

[cooljosh2k2]

This works for square roots, but not for cube roots, fourth roots, etc.

(a - b)(a + b) = a² - b²
If a is the square root of something, a² will have the radical eliminated.

(a - b)(a² + ab + b² = a³ - b³
This time, if a is the cube root of something, a³ will have the radical eliminated.

(a - b)(a³ +a²b + ab² + b³) = a⁴ - b⁴
In this example, if a is the fourth root of something, a⁴ will have the radical eliminated.

Notice that there is a pattern here.

Im confused, isn't that what i did? I removed the radical in the numerator by multiplying it by the the square root + 1, but it then leaves me with a radical in the denominator. I still don't understand how it would give me for a) a limit of 1/c.

 

[╔(σ_σ)╝]

Are you trying to prove or "evaluate" these limits because I don't see any epsilons and deltas.

 

[cooljosh2k2]

Are you trying to prove or "evaluate" these limits because I don't see any epsilons and deltas.

The question says "prove that...", but doesn't specify if i have to use epsilons and deltas or can use another theorem such as squeeze theorem, ratio theorem, etc to prove the limit.

 

[╔(σ_σ)╝]

Maybe you should ask your professor because what you are doing now is verifying not proving since you are already given the limits.

 

[Mark44]

This works for square roots, but not for cube roots, fourth roots, etc.

(a - b)(a + b) = a² - b²
If a is the square root of something, a² will have the radical eliminated.

(a - b)(a² + ab + b² = a³ - b³
This time, if a is the cube root of something, a³ will have the radical eliminated.

(a - b)(a³ +a²b + ab² + b³) = a⁴ - b⁴
In this example, if a is the fourth root of something, a⁴ will have the radical eliminated.

Notice that there is a pattern here.

Im confused, isn't that what i did? I removed the radical in the numerator by multiplying it by the the square root + 1, but it then leaves me with a radical in the denominator. I still don't understand how it would give me for a) a limit of 1/c.

You CANNOT remove the radical in the numerator if you multiply the "c"th root of 1 + x by the square root of 1 + x. Also, as ╔(σ_σ)╝ points out, you need to prove that the limits are as given, not just evaluate them. That will entail using an epsilon-delta proof.