Limits：練習問題

Evaluate the limit: \( \lim_{x \to \infty} \frac{6x^2 + 5x}{(2x + 1)(x - 3)} \)

Evaluate \( \lim_{x \to \infty} \left( 1 + \frac{2}{x} \right)^{3x} \)

Find the value of the limit: \( \lim_{x \to 0} \frac{\ln(1 + \sin(3x))}{x} \)

Evaluate the limit: \( \lim_{x \to \infty} \frac{\sqrt{x^2 + 1}}{3x - 2} \)

Evaluate the limit: \( \lim_{x \to 0} \frac{\sqrt{1+3x} - 1}{x^2 + 2x} \)

Evaluate the limit: \(\lim_{x \to -2} \frac{x^2 - 4}{x^2 + 5x + 6}\).

Evaluate the limit: \(\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}\).

Evaluate \( \lim_{x \to 0} \frac{\sin(2x)}{\tan(7x)} \).

Consider the function \( f(x) = \frac{\sqrt{4+x}-2}{x} \) for \( x \neq 0 \).
(a) Evaluate the limit \( \lim_{x \to 0} f(x) \).
(b) Using the result from (a), evaluate \( \lim_{x \to 0} \frac{\sqrt{4+\sin(3x)}-2}{\sin(3x)} \).

(a) Evaluate the limit \( \lim_{x \to 0} \frac{e^{2x} - \cos(x)}{x} \).
(b) Evaluate the limit \( \lim_{x \to \infty} (\sqrt{x^2 + 6x} - \sqrt{x^2 - x}) \).