Please provide the specific limits you'd like me to evaluate using algebraic simplification. I'd be happy to help!

Let me know if you want details or have any questions.
Here are some related examples of limits using algebraic simplification:

1. $\lim_{{x \to 0}} \frac{{\sin x}}{{x}}$
2. $\lim_{{x \to \infty}} \frac{{1}}{{x}}$
3. $\lim_{{x \to 1}} \frac{{x^2 - 1}}{{x - 1}}$
4. $\lim_{{x \to 0}} \frac{{x^2}}{{x}}$
5. $\lim_{{x \to \infty}} \frac{{x + 1}}{{x}}$

Tip: When simplifying limits, always check for indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, which can often be resolved using techniques like factoring or rationalizing the expression.