Explanation

To evaluate the limit, we can start by rewriting the expression as:

lim (sin(2x) + a*sin(x)) / x^3

As x approaches 0, the sine functions can be approximated by their arguments, so we get:

lim (2x + a*x) / x^3 = lim (2 + a) / x^2

For this limit to exist, the numerator must be 0, so we set 2 + a = 0, which gives a = -2.

Explanation

The equation z = a + 2by + 2bxy will be true when the terms containing b are zero, which happens when b = 0.

Explanation

Square root is a unary operation because it takes only one argument or input, whereas addition and multiplication are binary operations that require two inputs.

Explanation

• A subset F of a metric space X is closed if and only if it is equal to its closure (F = F°).
• This means that F contains all its limit points, and no other points.

Explanation

The power set of a nonempty set X is the set of all subsets of X, including X itself and the empty set. It forms a topology on X.

Explanation

To solve the inequality x−3/x+7 < 0, we need to find where the expression is negative.

We first determine the critical points by setting the numerator and denominator equal to zero:

x−3 = 0 gives x = 3, and x+7 = 0 gives x = −7.

Now, we create a sign chart with these critical points and test intervals to see where the expression is negative:

Test x = −8 (a number less than -7): −8−3/-8+7 = −11/-1 ​= 11>0 (positive)

Test x = 0 (a number between -7 and 3): 0−3/0+7 = −3/7 <0 (negative)

Test x = 4 (a number greater than 3): 4−3/ 4+7 = 1/11 >0 (positive)

From this, we see the expression is negative when x is between -7 and 3, so the solution is (−7,3).

Explanation

The correct answer is: Xn < X(n+1)

A sequence (Xn) in a metric space X is said to be strictly increasing if for all n, Xn is less than X(n+1), denoted by Xn < X(n+1). This means that each term of the sequence is strictly greater than the previous term.

Explanation

The set of nth roots of unity, denoted by {ω₀, ω₁, ..., ωₙ₋₁}, forms a group under the operation of multiplication.

Explanation

• A group of prime order (a group with a prime number of elements) is always a cyclic group.
• This is a fundamental result in group theory, known as Lagrange's theorem.

Explanation

• The set S = {1/n : n ∈ N} is a set of positive fractions, where n is a natural number.
• As n increases, the value of 1/n decreases and approaches 0.