1What kind of limit does \( \lim_{x \to a^-} f(x) \) represent: the right-hand limit, the left-hand limit, the value of the function, or none of these?
2What condition must be satisfied for the limit of \( f(x) \) to exist at \( x = c \)?
3What is the value of the function \( f(x) = 3x \) at \( x = 1 \)?
4Which condition is needed for a function \( f(x) \) to be continuous at \( x = c \)?
5If \( f(x) = x + 1 \), what is the value of \( \lim_{x \to 1} \frac{1}{f(x)} \)?
6What do \( \lim_{x \to c^-} f(x) \) and \( \lim_{x \to c^+} f(x) \) represent? Explain with an example.
7What are the three conditions needed for a function \( f(x) \) to be continuous at \( x = a \)?
8Let \( g(x) = x + 3 \). Find \( g(x) \) at \( x = 2.9, 2.99, 2.999 \) and at \( x = 3.1, 3.01, 3.001 \). Find \( g(3) \), the limits \( \lim_{x \to 3^+} g(x) \) and \( \lim_{x \to 3^-} g(x) \), and say whether \( g(x) \) is continuous at \( x = 3 \), with a reason.
9Let \( h(x) = 2x - 1 \). Find \( h(x) \) at \( x = 0.9, 0.99, 0.999 \) and at \( x = 1.1, 1.01, 1.001 \). Find \( h(1) \), the limits \( \lim_{x \to 1^+} h(x) \) and \( \lim_{x \to 1^-} h(x) \), and decide whether \( h(x) \) is continuous at \( x = 1 \).
10A function is given by \( k(x) = 3 - x \) for \( 0 \le x \le 1 \) and \( k(x) = x + 1 \) for \( x > 1 \). Find \( k(0.99) \) and \( k(1.01) \), find \( \lim_{x \to 1^-} k(x) \) and \( \lim_{x \to 1^+} k(x) \), and decide whether \( k(x) \) is continuous at \( x = 1 \).
11A function is given by \( p(x) = 2x + 1 \) for \( 1 \le x \le 3 \) and \( p(x) = x + 4 \) for \( x > 3 \). Find \( p(2.99) \) and \( p(3.01) \), find \( \lim_{x \to 3^-} p(x) \) and \( \lim_{x \to 3^+} p(x) \), and decide whether \( p(x) \) is continuous at \( x = 3 \).
12By putting in values of \( x \), test the continuity of \( f(x) = 2x + 5 \) at \( x = 1 \).
13By putting in values of \( x \), test the continuity of \( f(x) = \frac{x^2 - 4}{x - 2} \) at \( x = 2 \).
14By putting in values of \( x \), test the continuity of \( f(x) = \frac{x^3 - 27}{x - 3} \) at \( x = 3 \).
15A function is given by \( f(x) = 7x - 2 \) for \( x < 3 \), \( f(x) = 5 \) for \( x = 3 \), and \( f(x) = 6x + 1 \) for \( x > 3 \). Test the continuity of \( f(x) \) at \( x = 3 \).
16A function is given by \( g(x) = x + 2 \) for \( 0 \le x \le 1 \) and \( g(x) = 3 \) for \( x > 1 \). Test the continuity of the function at \( x = 1 \).
17A function is given by \( f(x) = 2x - 3 \) for \( x < 2 \), \( f(x) = 2k - 3 \) for \( x = 2 \), and \( f(x) = x + k \) for \( x > 2 \). What must the value of \( k \) be for \( f(x) \) to be continuous at \( x = 2 \)?
18The function \( g(x) = x + 3 \) is given. Explain, with an example, what it means for a curve to be continuous or discontinuous at a certain point. What does it mean to say that a curve is continuous from \( -4 \) to \( 4 \)? Is \( g(x) \) continuous at \( x = 3 \)? Give a reason.
19The function \( f(x) = 4x + 1 \) is defined. Calculate \( f(1.999) \) and \( f(2.001) \), find \( \lim_{x \to 2^-} f(x) \) and \( \lim_{x \to 2^+} f(x) \), calculate \( f(2) \), and say whether \( f(x) \) is continuous at \( x = 2 \), with a reason.
20The function \( p(x) = x + 5 \) for \( x \le 3 \) and \( p(x) = 4x - 4 \) for \( x > 3 \) is given. Write in symbols the condition for a function \( f(x) \) to be continuous at \( x = a \). Find \( p(2.9999) \) and \( p(3.0001) \), and say whether \( p(x) \) is continuous at \( x = 3 \), with a reason.
21A function \( h(x) = kx + 3 \) for \( x \le 1 \) and \( h(x) = 2x - 1 \) for \( x > 1 \) is given. What is a continuous function? Find the left-hand and right-hand limits at \( x = 1 \). If \( h(x) \) is continuous at \( x = 1 \), find the value of \( k \).
22If \( y = f(x) = x + 1 \), find \( f(2.999) \) and \( f(3.001) \). Is \( f(x) \) continuous at \( x = 3 \)? Give a reason.
23A function is given by \( f(x) = 4x + 3 \) for \( x \le 2 \) and \( f(x) = 5x + 1 \) for \( x > 2 \). Test the continuity of \( f(x) \) at \( x = 2 \).
24If \( g(x) = 5x + 2 \), find \( g(2.999) \) and \( g(3.001) \). Is \( g(x) \) continuous at \( x = 3 \)? Give a reason.
25The function \( f(x) = (x + 2)^2 + 5 \) is given. Sketch its graph. Is the graph continuous or discontinuous? Are the values of \( \lim_{x \to 3^-} f(x) \), \( \lim_{x \to 3^+} f(x) \) and \( f(3) \) equal? Give reasons.
26Test the continuity of \( f(x) = \frac{x^2 - 9}{x - 3} \) at \( x = 3 \).
27Can a function be discontinuous at \( x = c \) even though \( f(c) \) is defined? Give an example.
28A graph has a sharp V-shaped corner at \( x = 1 \) but no gap there. Is it continuous at \( x = 1 \)? Explain.