Limit notation and estimating limits from graphs and tables：練習題

Find the value of the constant \(k\) such that the function \(f(x) = \begin{cases} kx + 1 & x \le 3 \\ x^2 - 2 & x > 3 \end{cases}\) is continuous at \(x = 3\).

Consider the function \(g(x) = \frac{x^2 - 9}{|x - 3|}\). What is the value of \(\lim_{x \to 3^-} g(x)\)?

The graph of a function \(f(x)\) has a vertical asymptote at \(x = a\) and a removable discontinuity at \(x = b\). Which of the following could be the expression for \(f(x)\)?

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

Which of the following conditions is NOT required for a function \(f(x)\) to be continuous at \(x = c\)?

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

Find the value of the constant \(k\) such that the function \(f(x)\) is continuous at \(x = 3\):
\(f(x) = \begin{cases} \frac{x^2 - 9}{x - 3} & x \neq 3 \\ k & x = 3 \end{cases}\)

The graph of a function \(y = f(x)\) has a vertical asymptote at \(x = 1\) and a horizontal asymptote at \(y = 2\). If \(f(x) = \frac{ax + 3}{x - c}\), determine the values of the constants \(a\) and \(c\).

Given the piecewise function:
\( g(x) = \begin{cases} 2x + k & x < 2 \\ x^2 - 1 & x \geq 2 \end{cases} \)
(a) Find the value of the constant \( k \) such that \( g(x) \) is continuous at \( x = 2 \).
(b) Using the value of \( k \) found in part (a), find \( \lim_{x \to 2} g(x) \).

A function \( f \) is defined on the interval \([0, 4]\) such that:
\( f(x) = \frac{ax + b}{x^2 - 4x + 3} \)
(a) Identify the points of discontinuity of \( f(x) \) within the domain.
(b) If \( \lim_{x \to 1} f(x) \) exists and is equal to \( L \), find the relationship between constants \( a \) and \( b \).
(c) Given \( L = -2 \), find the specific values of \( a \) and \( b \).