Mathematics

This is a limit of an indeterminate form $\frac{0}{0}$, which signals that direct substitution won't work. The key insight is to rationalize the numerator by multiplying by the conjugate $\sqrt{1+3x}+1$—this eliminates the square root from the top and reveals a hidden factor of $x$ that cancels the denominator. Once the obstruction is cleared, the limit becomes straightforward to evaluate.

This is a limit of an indeterminate form $\frac{0}{0}$, which signals that direct substitution won't work. The key insight is to rationalize the numerator by multiplying by the conjugate $\sqrt{1+3x}+1$—this eliminates the square root from the top and reveals a hidden factor of $x$ that cancels the denominator. Once the obstruction is cleared, the limit becomes straightforward to evaluate.