Given:
We are given the limit as x approaches infinity of the expression:

lim(x -> ∞) 100[{(x+1)(x+2)(x+3)...(x+100)}^(1/100) - x]

Let's solve this limit step by step.

Step 1: Expanding the expression
We can expand the expression inside the curly brackets using the formula for the product of consecutive terms:

(x+1)(x+2)(x+3)...(x+100)

This can be written as:

x^100 + (1+2+3+...+100)x^99 + (1+2+3+...+100)(1+2+3+...+100)x^98 + ...

Simplifying further, we get:

x^100 + (1+2+3+...+100)x^99 + (1+2+3+...+100)^2x^98 + ...

Step 2: Simplifying the expression
We can see that as x approaches infinity, the terms with lower powers of x become insignificant compared to the highest power term (x^100). Therefore, we can ignore all the terms except for x^100.

Now, let's focus only on the x^100 term:

lim(x -> ∞) 100(x^100 - x)

Step 3: Simplifying the limit
We can factor out x from the expression inside the limit:

lim(x -> ∞) 100x(x^99 - 1)

As x approaches infinity, x^99 becomes much larger than 1. Therefore, we can ignore the -1 term compared to x^99.

Now, the expression becomes:

lim(x -> ∞) 100x(x^99)

Step 4: Evaluating the limit
As x approaches infinity, the term x^99 also approaches infinity. Therefore, we can rewrite the limit as:

∞ * ∞ * ∞ = ∞

So, the answer to the given limit is infinity, not 5050.

Note: The given answer of 5050 does not seem to be correct based on the provided expression and limit.