Question:

For $\lim_{x \to 0} \frac{x^2 + 5x}{x^2 + x^3}$, factor the numerator and denominator: $\lim_{x \to 0} \frac{x(x + 5)}{x^2(1 + x)} = \lim_{x \to 0} \frac{x + 5}{x(1 + x)}$. As $x \to 0$, the numerator approaches $5$ while the denominator approaches $0$, making this limit infinite; however, checking the original form by factoring out $x$: $\lim_{x \to 0} \frac{x(x+5)}{x^2(1+x)} = \lim_{x \to 0} \frac{(x+5)}{x(1+x)}$ yields $\frac{5}{0} = \infty$ or recomputing gives $5$ with proper simplification.