Using L'Hôpital's rule, the limit evaluates to 5.

The limit lim(x→0) (sin(x)/x) = 1.

As x approaches infinity, 1/x approaches 0.

Substituting x = 2 into the polynomial gives p(2) = 3(2^2) - 2(2) + 1 = 12 - 4 + 1 = 9.

Substituting x = 2 into the polynomial gives p(2) = 3(2^2) - 4(2) + 1 = 12 - 8 + 1 = 5.

Substituting x = 2 into the polynomial gives P(2) = 4(2^2) - 3(2) + 7 = 16 - 6 + 7 = 27.

Substituting x = -1 gives P(-1) = 5(-1)^2 - 3(-1) + 7 = 5 + 3 + 7 = 15.

The principal quantum number n for the f subshell is 4.

The principal quantum number (n) indicates the energy level of the electron. For 4s, n=4.

In the ground state, the electron is in the first energy level, which corresponds to n=1.

Sodium has an atomic number of 11, and its electron configuration is 1s² 2s² 2p⁶ 3s¹. The outermost electrons are in n=3.

Potassium has the electronic configuration [Ar] 4s1, so the outermost electron is in n=4.

Potassium (K) has an atomic number of 19, and its electron configuration ends in 4s, so the outermost electron has n=4.

The ground state of hydrogen corresponds to n=1.

Sodium has an atomic number of 11, and its outermost electrons are in the n=3 shell.

Potassium has an atomic number of 19, and its outermost electrons are in the n=4 shell.

The series diverges to infinity as n approaches infinity.

The square root of 144 is 12, since 12 * 12 = 144.

The term containing x^4 is given by C(6, 4)(3x)^4(4)^2 = 15 * 81 * 16 = 19440.

The term containing x^4 is C(8,4) * (1/2)^4 = 70 * 1/16 = 4.375.

The term containing x^5 is C(8,5)(1/2)^3 = 56 * 1/8 = 7.

The term containing x^5 is C(8,5)(2)^3 = 56 * 8 = 448.

The universal gas constant R is 8.314 J/(mol·K).

The universal gas constant R is 0.0821 L·atm/(K·mol).

Subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4.

Subtracting 3 from both sides gives 2x = 12. Dividing by 2 gives x = 6.

Subtracting 5 from both sides gives 3x = 15. Dividing by 3 gives x = 5.

Solving for x: 3x = 21 => x = 7.

Rearranging gives 3x - 2x = 5 + 7, thus x = 12.

A matrix is singular if its determinant is zero. Det(A) = (x*4) - (2*3) = 4x - 6 = 0. Solving gives x = 1.5.