The limit is 0.

Using the identity 1 - cos(θ) = 2sin^2(θ/2), we have lim (x -> 0) (1 - cos(2x))/x^2 = lim (x -> 0) (2sin^2(x))/x^2 = 2.

Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(2x))/(x^2) = 8.

Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1.

Using the Taylor series expansion for cos(x), we find that lim (x -> 0) (cos(x) - 1)/x^2 = -1/2.

Using the derivative of e^x at x = 0, we find lim (x -> 0) (e^x - 1)/x = 1.

Using the limit property, lim (x -> 0) (sin(kx)/x) = k. Here, k = 5, so the limit is 5.

Since |sin(1/x)| <= 1, we have |x^2 * sin(1/x)| <= |x^2|. As x approaches 0, |x^2| approaches 0, hence the limit is 0.

As x approaches 0, e^x - 1 approaches 0. Using L'Hôpital's Rule three times, we find the limit approaches 0.

This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1) = x + 1. Thus, lim (x -> 1) (x + 1) = 2.

Factoring gives (x - 1)(x + 1)/(x - 1)^2. Canceling gives lim (x -> 1) (x + 1)/(x - 1) = 2.

Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^2 + x + 1) = 3.

The expression (x^2 - 4)/(x - 2) can be factored as (x - 2)(x + 2)/(x - 2). Canceling (x - 2) gives lim (x -> 2) (x + 2) = 4.

Dividing numerator and denominator by x^2, we get lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.

Using the limit property, lim(x->0) (tan(kx)/x) = k. Here, k = 3, so the limit is 3.

Magnitude = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Magnitude |v| = √(3^2 + (-4)^2 + 12^2) = √(9 + 16 + 144) = √169 = 13.

The maximum occurs at x = 2. f(2) = -2^2 + 4(2) + 1 = 5.

The vertex form gives maximum at x = 2. f(2) = -2^2 + 4*2 = 4.

The function is a downward-opening parabola. The maximum occurs at x = -b/(2a) = 8/(2*2) = 2. f(2) = -2(2^2) + 8(2) - 3 = 8.

The function is a downward-opening parabola. The maximum occurs at x = -b/(2a) = 8/(2*2) = 2. f(2) = -2(2^2) + 8(2) - 5 = 9.

The vertex occurs at x = 2. f(2) = -2^2 + 4(2) + 1 = 9, which is the maximum value.

The vertex occurs at x = 3. f(3) = -3^2 + 6(3) - 8 = 6.

Arranging the numbers: 1, 3, 3, 6, 7, 8, 9. Median = 6 (middle value).

Arranging the numbers: 1, 3, 3, 6, 7, 8, 9. Median = 6 (middle value).

Arranging the numbers: 1, 2, 3, 4. Median = (2 + 3) / 2 = 2.5.

Midpoint = ((1+3)/2, (2+4)/2) = (2, 3).

Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 4)/2, (3 + 7)/2) = (3, 5).

The vertex occurs at x = -b/(2a) = 12/6 = 2. f(2) = 3(2^2) - 12(2) + 7 = 1.

The vertex of the parabola occurs at x = 2. f(2) = 2^2 - 4(2) + 5 = 1, which is the minimum value.