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 11
A function f(x) is continuous everywhere except at x = 2, where it has a jump. What type of discontinuity is this?
Evaluate \( \lim_{x \to \infty} \frac{2x^2 + 3}{x^2 - 1} \).
Evaluate \( \lim_{x \to 0} \frac{\sin x}{x^2} \).
Evaluate \( \lim_{x \to 0} \frac{\sin x}{x} \).
Evaluate \( \lim_{x \to 0} \frac{e^x - 1}{x} \).
Evaluate \( \lim_{x \to 1} \frac{x^2 - 1}{x - 1} \).
Evaluate \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \).
A function f(x) is continuous everywhere except at x = 2, where it has a jump. What type of discontinuity is this?
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 removable discontinuity?
(1) f(x) = \frac{x^2 - 1}{x - 1}
(2) f(x) = \frac{1}{x}
(3) f(x) = \tan x
(4) f(x) = |x|
Which of the following is an example of a function with a jump discontinuity?
(1) f(x) = |x|
(2) f(x) = \frac{1}{x}
(3) f(x) = \begin{cases} 1, & x < 0 \\ 2, & x \geq 0 \end{cases}
(4) f(x) = x^2
Which of the following is NOT a property of limits?
(1) \( \lim_{x \to a} [cf(x)] = c \lim_{x \to a} f(x) \)
(2) \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \)
(3) \( \lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \times \lim_{x \to a} g(x) \)
(4) \( \lim_{x \to a} [f(x)/g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \)
\( \lim_{x \to 0} \frac{\sin 3x}{x} = \underline{\hspace{2cm}} \)
\( \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \underline{\hspace{2cm}} \)
A function f(x) is said to be ______ at x = a if \( \lim_{x \to a} f(x) = f(a) \).
A function is said to have an infinite discontinuity at x = a if ______.
If \( \lim_{x \to 1^-} f(x) = 2 \) and \( \lim_{x \to 1^+} f(x) = 3 \), does \( \lim_{x \to 1} f(x) \) exist?
If \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \), then \( \lim_{x \to a} f(x) = L \). True or False?
True or False: If a function is continuous at x = a, then \( \lim_{x \to a} f(x) \) exists.
True or False: If a function is continuous at x = a, then it must be defined at x = a.
