Limit notation and estimating limits from graphs and tables：練習問題

Evaluate the following limit:
\(\lim_{x \to \infty} \frac{4x^2 - 3x + 7}{2x^2 + 5x - 1}\)

Consider the function \(f(x)\) defined by:
\(f(x) = \begin{cases} \frac{e^{3x} - 1}{x} & x < 0 \\ a & x = 0 \\ b \cos(x) + \frac{\sin(2x)}{x} & x > 0 \end{cases} \)
If \(f(x)\) is continuous at \(x = 0\), find the values of the constants \(a\) and \(b\).

Evaluate the limit:
\(\lim_{x \to 0^+} (\cos(x))^{1/x^2}\)

Evaluate the limit:
\(\lim_{x \to 0} \frac{1 - \cos(4x)}{x^2}\)

Consider the function \(g(x)\) defined by:
\(g(x) = \begin{cases} \frac{k \sin(x)}{x} & x < 0 \\ 4 & x = 0 \\ 2x + m & x > 0 \end{cases} \)
If \(g(x)\) is continuous at \(x = 0\), find the values of the constants \(k\) and \(m\).

Evaluate the limit:
\(\lim_{x \to \infty} \frac{3x^2 - 5x + 2}{7x^2 + 1}\)

Let \(f(x) = \frac{|x-3|}{x-3}\). Find the value of \(\lim_{x \to 3^+} f(x)\) and determine if \(\lim_{x \to 3} f(x)\) exists.

Find the value of the constant \(k\) such that the limit \(\lim_{x \to 0} \frac{\sin(kx) \tan(2x)}{x^2} = 8\).

Consider the function \( f(x) \) defined by:
\( f(x) = \begin{cases} \frac{1 - \cos(kx)}{x^2} & x < 0 \\ A & x = 0 \\ \frac{\sqrt{1 + mx} - \sqrt{1 - mx}}{x} & x > 0 \end{cases} \)
where \( k \), \( m \), and \( A \) are constants.

(a) Find the limit \( \lim_{x \to 0^-} f(x) \) in terms of \( k \).
(b) Find the limit \( \lim_{x \to 0^+} f(x) \) in terms of \( m \).
(c) If \( f(x) \) is continuous at \( x = 0 \), express \( k \) in terms of \( m \).
(d) Given that \( f(x) \) is continuous at \( x = 0 \) and \( A = 2 \), find the possible values of \( k \) and \( m \).

Consider the function \(f(x)\) defined by:
\(f(x) = \begin{cases} \frac{\sin(ax)}{x} + b & \text{if } x < 0 \\ 3 & \text{if } x = 0 \\ \frac{\sqrt{1 + cx} - 1}{x} & \text{if } x > 0 \end{cases}\)
where \(a, b, \text{ and } c\) are constants.

(a) Find the value of \(c\) such that \(\lim_{x \to 0^+} f(x) = f(0)\).
(b) Find a relationship between \(a\) and \(b\) such that \(f(x)\) is continuous at \(x = 0\).
(c) If it is further given that the slope of the tangent to the curve \(y = \frac{\sin(ax)}{x} + b\) approaches \(2\) as \(x\) approaches \(0^-\), find the specific values of \(a\) and \(b\). (Note: Use the Taylor expansion or L'Hôpital's Rule for the derivative limit if necessary).