`lim_(xtooo) ((x+1)^(10)+(x+2)^(10)+...+(x+100)^(10))/(x^(10)+10^(10))` is equal to

Options

A. 0
B. 1
C. 2
D. 4

Explanation

To solve the limit

\[ \lim_{x \to \infty} \frac{(x+1)^{10} + (x+2)^{10} + \ldots + (x+100)^{10}}{x^{10} + 10^{10}}, \]

we can follow these steps:

### Step 1: Analyze the limit
As \( x \) approaches infinity, both the numerator and denominator approach infinity. Thus, we can simplify the expression by factoring out the highest power of \( x \) from both the numerator and the denominator.

### Step 2: Factor out \( x^{10} \)
In the numerator, we can factor out \( x^{10} \) from each term:

\[ (x+1)^{10} = x^{10} \left(1 + \frac{1}{x}\right)^{10}, \quad (x+2)^{10} = x^{10} \left(1 + \frac{2}{x}\right)^{10}, \ldots, \quad (x+100)^{10} = x^{10} \left(1 + \frac{100}{x}\right)^{10}. \]

Thus, the numerator becomes:

\[ x^{10} \left( \left(1 + \frac{1}{x}\right)^{10} + \left(1 + \frac{2}{x}\right)^{10} + \ldots + \left(1 + \frac{100}{x}\right)^{10} \right). \]

In the denominator, we can also factor out \( x^{10} \):

\[ x^{10} + 10^{10} = x^{10} \left(1 + \frac{10^{10}}{x^{10}}\right). \]

### Step 3: Rewrite the limit
Now we can rewrite the limit:

\[ \lim_{x \to \infty} \frac{x^{10} \left( \left(1 + \frac{1}{x}\right)^{10} + \left(1 + \frac{2}{x}\right)^{10} + \ldots + \left(1 + \frac{100}{x}\right)^{10} \right)}{x^{10} \left(1 + \frac{10^{10}}{x^{10}}\right)}. \]

### Step 4: Cancel \( x^{10} \)
We can cancel \( x^{10} \) from the numerator and the denominator:

\[ \lim_{x \to \infty} \frac{\left(1 + \frac{1}{x}\right)^{10} + \left(1 + \frac{2}{x}\right)^{10} + \ldots + \left(1 + \frac{100}{x}\right)^{10}}{1 + \frac{10^{10}}{x^{10}}}. \]

### Step 5: Evaluate the limit
As \( x \to \infty \), \( \frac{k}{x} \to 0 \) for any constant \( k \). Therefore, each term \( \left(1 + \frac{k}{x}\right)^{10} \) approaches \( 1^{10} = 1 \).

There are 100 terms in the numerator, each approaching 1:

\[ \left(1 + \frac{1}{x}\right)^{10} + \left(1 + \frac{2}{x}\right)^{10} + \ldots + \left(1 + \frac{100}{x}\right)^{10} \to 1 + 1 + \ldots + 1 = 100. \]

In the denominator, \( \frac{10^{10}}{x^{10}} \to 0 \), so:

\[ 1 + \frac{10^{10}}{x^{10}} \to 1. \]

### Step 6: Final result
Thus, we have:

\[ \lim_{x \to \infty} \frac{100}{1} = 100. \]

So, the final answer is:

\[ \boxed{100}. \]