Explanation

Given lim x → 1⁺ (2 + √x - 1)

As x approaches 1 from the right:

√x approaches √1 = 1

So, √x - 1 approaches 1 - 1 = 0

2 + 0 = 2

The limit is 2.

Explanation

Given f(x) = 6x/(3x - 2)

The denominator cannot be zero:

3x - 2 ≠ 0

3x ≠ 2

x ≠ 2/3

So, the domain of f(x) is all real numbers except x = 2/3.

The answer is R - {2/3}.

Explanation

Given the conditions:

1. √x requires x ≥ 1 (from x - 1 ≥ 0)

2. Denominator x² - 2x ≠ 0 implies x ≠ 0 and x ≠ 2

The domain is indeed x ≥ 1, excluding x = 2.

So, the domain is (1, 2) ∪ (2, ∞).

Explanation

Given f(t) = 1/t and g(t) = t³

To find gof(t), substitute f(t) into g(t):

gof(t) = g(f(t))

= g(1/t)

= (1/t)³

= 1/t³

Explanation

The differential equation (1 - x²)y'' - 2xy' + n(n + 1)y = 0 is known as Legendre's equation.

It arises in problems with spherical symmetry, such as solutions to Laplace’s equation in spherical coordinates.

Explanation

Given a ≠ 0, b ≠ 0, and ab = 0:

In this case, a and b are called Zero divisors.

Explanation

Given:  

5 - x = 25

Step 1: Subtract 5 from both sides  

-x = 25 - 5  

-x = 20

Step 2: Multiply both sides by -1  

x = -20

Explanation

Given tan θ < 0 and cos θ > 0:

Since cos θ > 0, θ lies in either the 1st or 4th quadrant.

Given tan θ < 0, θ lies in either the 2nd or 4th quadrant.

Combining both conditions, θ must lie in the 4th quadrant.

Explanation

Given 2 is a zero of the polynomial 2x⁴ + x³ + kx² - 8, we substitute x = 2 into the polynomial and set it equal to 0:

2(2)⁴ + (2)³ + k(2)² - 8 = 0

2(16) + 8 + 4k - 8 = 0

32 + 8 + 4k - 8 = 0

32 + 4k = 0

Now, solve for k:

4k = -32

k = -32/4

k = -8

Explanation

Slope AB = (-2-2)/(5-1) = -4/4 = -1

Slope BC = (2-5)/(1-(-2)) = -3/3 = -1

Since slopes AB and BC are equal (-1), the points A, B, and C are collinear.