Limits and Continuity
Math ⇒ Calculus
Limits and Continuity starts at 11 and continues till grade 12.
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See sample questions for grade 12
Evaluate \( \lim_{x \to 0} \frac{\ln(1 + x)}{x} \).
Evaluate \( \lim_{x \to 0} \frac{\sin 3x}{x} \).
Evaluate \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} \).
Evaluate \( \lim_{x \to 0} \frac{e^x - 1}{x} \).
Evaluate \( \lim_{x \to 1} \frac{x^2 - 1}{x - 1} \).
If \( \lim_{x \to 0} f(x) = 2 \) and \( \lim_{x \to 0} g(x) = -1 \), find \( \lim_{x \to 0} [2f(x) - 3g(x)] \).
If \( \lim_{x \to 3} f(x) = 7 \) and \( \lim_{x \to 3} g(x) = 2 \), find \( \lim_{x \to 3} [f(x) + g(x)] \).
If \( f(x) = \begin{cases} 2x + 1 & x < 1 \\ 3 & x = 1 \\ x^2 & x > 1 \end{cases} \), is f(x) continuous at x = 1?
If \( f(x) = \begin{cases} x^2 & x \neq 2 \\ 5 & x = 2 \end{cases} \), is f(x) continuous at x = 2?
If \( \lim_{x \to a} f(x) \) does not exist, which of the following could be a reason?
(1) The left-hand and right-hand limits are not equal.
(2) The function approaches infinity.
(3) The function oscillates as x approaches a.
(4) All of the above.
Which of the following functions is not continuous at x = 0?
(1) \( f(x) = x^2 \)
(2) \( f(x) = \frac{1}{x} \)
(3) \( f(x) = \sin x \)
(4) \( f(x) = e^x \)
Which of the following is a necessary condition for a function to be continuous at x = a?
(1) \( \lim_{x \to a} f(x) \) exists
(2) f(a) is defined
(3) \( \lim_{x \to a} f(x) = f(a) \)
(4) All of the above
Which of the following is an example of an infinite discontinuity?
(1) \( f(x) = \frac{1}{x} \) at x = 0
(2) \( f(x) = x^2 \) at x = 0
(3) \( f(x) = \sin x \) at x = 0
(4) \( f(x) = |x| \) at x = 0
A function is said to be continuous at a point x = a if ________.
Fill in the blank: The function \( f(x) = \tan x \) is discontinuous at x = ________.
If \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), then \( \lim_{x \to a} [f(x)g(x)] = \) ________.
If \( f(x) = \begin{cases} 2x + 1 & x < 1 \\ 3 & x = 1 \\ x^2 & x > 1 \end{cases} \), is f(x) continuous at x = 1?
If \( f(x) = \begin{cases} x^2 & x \neq 2 \\ 5 & x = 2 \end{cases} \), is f(x) continuous at x = 2?
True or False: Every rational function is continuous everywhere in its domain.
True or False: If \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \), then \( \lim_{x \to a} f(x) \) exists and equals L.
