ers a) \( \operatorname{Lim}_{x \rightarrow 2} \frac{7 x^{3}-10 x^{4}-13 x+6}{3 x^{2}-6 x-8}= \) b) \( \lim _{x \rightarrow 1} \frac{x^{3}+3 x+7}{x^{2}-2 x+2}= \) f) \( \lim _{x \rightarrow 2} \frac{\frac{3}{2} x^{3}-\frac{1}{2} x^{2}+\frac{1}{3} x+\frac{1}{4}}{-\frac{3}{4} x^{2}+\frac{5}{3} x+\frac{5}{4}}= \) c) \( \lim _{x \rightarrow 1} \frac{-3 x^{3}+2 x^{2}-x+3}{-8 x^{2}-6 x+36}= \) g) \( \lim _{x \rightarrow \frac{1}{2}} \frac{\frac{1}{2} x^{2}-\frac{1}{3} x+\frac{2}{5}}{\frac{1}{3} x^{2}+\frac{1}{2} x+\frac{3}{4}}= \) d) \( \lim _{x \rightarrow 3} \frac{\sqrt{4 x^{2}+5}-x^{2}}{\left(x^{3}+5\right)^{2}+3 x-5}= \) e) \( \lim _{x \rightarrow 1} \frac{\sqrt{3 x}+3 x^{2}+2 x}{\sqrt[3 x^{2}]{ }-4 x^{3}-6}= \) h) \( \lim _{x \rightarrow \frac{1}{2}} \frac{-\frac{3}{4} x^{3}+\frac{2}{3} x^{2}-\frac{1}{4} x+\frac{5}{3}}{\frac{4}{3} x^{2}+\frac{5}{2} x-\frac{1}{2}}= \)

Calculating limits can often seem daunting, but there's a fun historical tidbit associated with it! The concept of limits was formalized in the 19th century by mathematicians like Augustin-Louis Cauchy and Karl Weierstrass. Their work laid down the foundational principles of calculus, allowing future generations to tackle problems involving continuous functions and their behaviors as they approach certain points. It's like the ancient Greeks inventing the wheel — the simple concept paved the way for the vast world of calculus we enjoy today! If you're diving into the realm of limits, remember that simplifying expressions is your best friend. Before plugging in the x-value, always look for common factors or opportunities to cancel out terms (without breaking the rules, of course!). A common mistake is to hastily substitute the value in and be shocked by getting a \( \frac{0}{0} \) indeterminate form. Keep your eyes peeled for algebraic manipulations like factoring or rationalizing, which can often help unravel the mystery of an elusive limit!