Question

$\lim_{x \to \infty} \frac{(2x+1)^{50} + (2x+2)^{50} + (2x+3)^{50} + \dots + (2x+100)^{50}}{(2x)^{50} + (10)^{50}} = $ ______.

• A. 50
• B. 100
• C. 25
• D. 200

Hint:

For $x \to \infty$ limits, ALWAYS ignore everything except the highest power of $x$. If the degrees are equal, the answer is just the ratio of their leading coefficients!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are asked to evaluate the limit of a massive rational polynomial function as $x \to \infty$.

Step 2: Detailed Explanation:
When taking the limit of a polynomial fraction as $x$ approaches infinity, only the highest degree terms (the leading terms) in the numerator and denominator dictate the final value. Lower degree terms become infinitesimally small in comparison and can be ignored.
First, determine the leading term of the denominator:
The denominator is $(2x)^{50} + 10^{50}$.
As $x \to \infty$, the constant $10^{50}$ is negligible.
Leading term of denominator = $(2x)^{50} = 2^{50} x^{50}$.
Next, determine the leading term of the numerator:
The numerator consists of a sum of exactly 100 binomial terms (from $+1$ to $+100$).
$(2x+1)^{50} + (2x+2)^{50} + \dots + (2x+100)^{50}$
If we expand each binomial using the Binomial Theorem, the highest power of $x$ in every single term will be $(2x)^{50}$. All other terms will be $x^{49}$, $x^{48}$, etc., which are negligible.
Since there are exactly 100 of these binomial terms, we add the leading term $(2x)^{50}$ exactly 100 times.
Leading term of numerator = $100 \times (2x)^{50} = 100 \times 2^{50} x^{50}$.
Now, divide the leading term of the numerator by the leading term of the denominator:
$\text{Limit} = \frac{100 \times 2^{50} x^{50}}{2^{50} x^{50}}$
The $2^{50} x^{50}$ perfectly cancels out:
$\text{Limit} = 100$

Step 3: Final Answer:
The limit is 100, matching option (b).