Continuity practice · Easy

Option C ($f(x)$ is discontinuous at $x=0$) is correct because the right-hand limit at x=0 is infinity, which is not equal to the function's value at x=0.

Option D ($\ln \left(\frac{2}{\mathrm{e}}\right)$) is correct because it is the limit of the function as x approaches 0, making the function continuous.

The correct answer is C ($\left( {2n + 1} \right)\frac{\pi }{{2}},n \in I$) because the tangent function is undefined at odd multiples of $\frac{\pi}{2}$.

Option B (-$\frac{1}{2}$) is correct because it is the value of p that makes the function continuous at x=0.

Option D (1) is correct because for the function to be continuous at x=3, the left-hand limit must equal the function's value at x=3.

Option D (1/4) is correct because the limit of the function as x approaches pi/4 is found using L'Hopital's rule, and for continuity, this limit must equal a.

Option B ($f$ is continuous at x=4) is correct because the left-hand limit, right-hand limit, and the function value at x=4 are all equal.

Option C (e) is correct because the limit of (x+1)^{(cot x)} as x approaches 0 is e, which is the value of f(0) for continuity.

Option C (1) is correct because for the function to be continuous at x=3, the left-hand limit must equal the function's value at x=3.

Option B (f is not continuous at $x=0$ for any real $k$) is correct because the limit as $x$ approaches 0 does not exist.

Option B (-1) is correct because the continuity of the function at x=1 and x=-1 leads to the equation a+b=-1.

Option D (103) is correct because the greatest integer function is discontinuous at every integer value, and there are 103 integers within the specified interval.

Option B ($f(x)$ is discontinuous at $x=a$) is correct because the left-hand limit and right-hand limit are not equal at $x=a$.

Option C (0) is correct because the limit of the function as x approaches $\pi/2$ is 0, which must equal $\lambda$ for continuity.

Option A (2) is incorrect. The function f(x) = [x] is discontinuous at x = 8 within the interval [7, 9].

The correct answer is D ($a = 1, b = -1$) because for the function to be continuous at x=4, the left-hand limit, right-hand limit, and the function's value at x=4 must be equal.

Option D ($\sqrt{5}-3$) is correct because by factoring the given equation and using the continuity of f(x), we can find f($\sqrt{5}$).

Option B (6) is correct because the limit of the function as x approaches pi/2 must equal the function's value at pi/2 for continuity.

Option A (3) is correct because for the function to be continuous at x=0, the limit as x approaches 0 must equal the function's value at x=0, which is k.

Option B (Discontinuous at only one point) is correct because the function is discontinuous only at x=0 after evaluating the limits.

Option B is correct because the function is undefined when the greatest integer function [x] equals zero, which occurs in the interval (0, 1), and it's discontinuous at all integers due to jumps in the greatest integer function.

Option C (3) is correct because the limit of the function as x approaches 0 must equal the function's value at x=0 for continuity, leading to k=3.

Option A is correct because the left-hand limit and right-hand limit at x=π/2 are not equal, indicating a discontinuity.

Option D ($a=-1, b=1$) is correct because these values ensure the function is continuous at x=1 and x=$\sqrt{2}$.

Option C (imaginary) is correct because the continuous function assumes only rational values, implying it's a constant function, leading to a quadratic equation with imaginary roots.

Option D (not continuous for any values of a and b) is correct because the function cannot be made continuous at all points simultaneously for any values of a and b.

Option D (1) is correct because for the function to be continuous at x=1, the left-hand limit must equal the right-hand limit, allowing us to solve for a.

Option B (13) is correct because there are 13 points of discontinuity of the function $f(x) = [2x^3 - 5]$ in the interval $[1, 2)$.

Option C (${e^3}$) is correct because the limit of the given expression as x approaches 0 is equal to ${e^3}$.

The correct answer is A ((-3 / 2,1)) because the points (α, β) and (a, a²) lie on the same side of the line x+2y-3=0, leading to the inequality (2a + 3)(a - 1) < 0.