Using L'Hôpital's Rule, we differentiate the numerator and denominator to find lim (x -> 0) (1/(1 + x)) = 1.

Since |sin(1/x)| ≤ 1, we have |x^2 sin(1/x)| ≤ |x^2|. Thus, lim (x -> 0) x^2 sin(1/x) = 0.

Using the fact that sin(x) ~ x as x approaches 0, we find that lim (x -> 0) (x^3)/(sin(x)) = 0.

Factoring gives lim (x -> 2) ((x - 2)(x^2 + 2x + 4))/(x - 2) = lim (x -> 2) (x^2 + 2x + 4) = 12.

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

Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x + 1/x^2) = 3/5.

Using the fact that sin(x) approaches x as x approaches 0, we have lim (x -> 0) (x^3)/(sin(x)) = 0.

As x approaches infinity, the leading terms dominate. Thus, lim (x -> ∞) (3x^2)/(5x^2) = 3/5.

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

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 that lim (x -> 0) (e^x - 1)/x = 1.

Using L'Hôpital's Rule, we differentiate the numerator and denominator to find lim (x -> 0) (1/(1 + x)) = 1.

Using the limit lim (x -> 0) (tan(x)/x) = 1, we have lim (x -> 0) (tan(3x)/x) = 3 * lim (x -> 0) (tan(3x)/(3x)) = 3.

Factoring gives (x - 1)(x + 1)/(x - 1), which simplifies to x + 1. Thus, lim (x -> 1) (x + 1) = 2.