Step 1: Understanding the Concept:
For limits at infinity involving polynomials, the result is determined by the ratio of the coefficients of the highest power of $x$.
Step 2: Formula Application:
Divide both numerator and denominator by the highest power of $x$, which is $x^{50}$.
Step 3: Explanation:
In the numerator, each term is of the form $(2x+k)^{50}$. The coefficient of $x^{50}$ in each of the 100 terms is $2^{50}$.
Total coefficient of $x^{50}$ in numerator = $100 \times 2^{50}$.
In the denominator, the term $(2x)^{50}$ has the coefficient $2^{50}$ for $x^{50}$.
Limit = $\frac{\text{Coeff of } x^{50} \text{ in Num}}{\text{Coeff of } x^{50} \text{ in Den}} = \frac{100 \times 2^{50}}{2^{50}} = 100$.
Step 4: Final Answer:
The limit is 100.