$lim_{x o c} [f(x) + g(x)] = lim_{x o c} f(x) + lim_{x o c} g(x)$

$lim_{x o c} [f(x) - g(x)] = lim_{x o c} f(x) - lim_{x o c} g(x)$

$lim_{x o c} [k cdot f(x)] = k cdot lim_{x o c} f(x)$

$lim_{x o c} [f(x) cdot g(x)] = lim_{x o c} f(x) cdot lim_{x o c} g(x)$

$lim_{x o c} \frac{f(x)}{g(x)} = \frac{lim_{x o c} f(x)}{lim_{x o c} g(x)}$, if $lim_{x o c} g(x) eq 0$

$lim_{x o c} [f(x)]^n = [lim_{x o c} f(x)]^n$

$lim_{x o c} \sqrt[n]{f(x)} = \sqrt[n]{lim_{x o c} f(x)}$

$lim_{x \to c} k = k$

$lim_{x \to c} x = c$

$lim_{x \to c} x^n = c^n$

The value that a function approaches as the input approaches some value.

Evaluating a limit by plugging in the value that x is approaching into the function.

An expression whose value cannot be determined, such as 0/0.

The limit of a sum is the sum of the limits: $lim_{x o c} [f(x) + g(x)] = lim_{x o c} f(x) + lim_{x o c} g(x)$

The limit of a difference is the difference of the limits: $lim_{x o c} [f(x) - g(x)] = lim_{x o c} f(x) - lim_{x o c} g(x)$

The limit of a constant times a function is the constant times the limit of the function: $lim_{x o c} [k cdot f(x)] = k cdot lim_{x o c} f(x)$

The limit of a product is the product of the limits: $lim_{x o c} [f(x) cdot g(x)] = lim_{x o c} f(x) cdot lim_{x o c} g(x)$

The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero: $lim_{x o c} \frac{f(x)}{g(x)} = \frac{lim_{x o c} f(x)}{lim_{x o c} g(x)}$, if $lim_{x o c} g(x) eq 0$

The limit of a function raised to a power is the limit of the function raised to that power: $lim_{x o c} [f(x)]^n = [lim_{x o c} f(x)]^n$

The limit of a root of a function is the root of the limit of the function: $lim_{x o c} \sqrt[n]{f(x)} = \sqrt[n]{lim_{x o c} f(x)}$

1. Direct substitution: $(2)^2 + 3(2) - 5 = 4 + 6 - 5 = 5$. Therefore, the limit is 5.

1. Direct substitution yields 0/0. 2. Factor the numerator: $x^2 - 9 = (x - 3)(x + 3)$. 3. Simplify: $\frac{(x - 3)(x + 3)}{x - 3} = x + 3$. 4. Evaluate the limit: $lim_{x o 3} (x + 3) = 3 + 3 = 6$.

1. Direct substitution yields 0/0. 2. Rationalize the numerator by multiplying by the conjugate: $\frac{\sqrt{x + 4} - 2}{x} cdot \frac{\sqrt{x + 4} + 2}{\sqrt{x + 4} + 2}$. 3. Simplify: $\frac{x + 4 - 4}{x(\sqrt{x + 4} + 2)} = \frac{x}{x(\sqrt{x + 4} + 2)} = \frac{1}{\sqrt{x + 4} + 2}$. 4. Evaluate the limit: $lim_{x o 0} \frac{1}{\sqrt{x + 4} + 2} = \frac{1}{\sqrt{4} + 2} = \frac{1}{4}$.

1. Direct substitution: $2(5)^2 - 3 = 2(25) - 3 = 50 - 3 = 47$. Therefore, the limit is 47.

1. Direct substitution yields 0/0. 2. Factor the numerator: $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$. 3. Simplify: $\frac{(x + 2)(x^2 - 2x + 4)}{x + 2} = x^2 - 2x + 4$. 4. Evaluate the limit: $lim_{x o -2} (x^2 - 2x + 4) = (-2)^2 - 2(-2) + 4 = 4 + 4 + 4 = 12$.

1. Direct substitution yields 0/0. 2. Rationalize the numerator: $\frac{\sqrt{x} - 2}{x - 4} cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2}$. 3. Simplify: $\frac{x - 4}{(x - 4)(\sqrt{x} + 2)} = \frac{1}{\sqrt{x} + 2}$. 4. Evaluate the limit: $lim_{x o 4} \frac{1}{\sqrt{x} + 2} = \frac{1}{\sqrt{4} + 2} = \frac{1}{4}$.

1. Direct substitution yields 0/0. 2. Factor the numerator: $x^2 + 4x - 5 = (x - 1)(x + 5)$. 3. Simplify: $\frac{(x - 1)(x + 5)}{x - 1} = x + 5$. 4. Evaluate the limit: $lim_{x o 1} (x + 5) = 1 + 5 = 6$.

1. Direct substitution yields 0/0. 2. Expand: $\frac{x^2+10x+25 - 25}{x} = \frac{x^2+10x}{x}$. 3. Simplify: $\frac{x(x+10)}{x} = x+10$. 4. Evaluate the limit: $lim_{x \to 0} (x+10) = 0+10 = 10$.

1. Direct substitution: $5(2) - 3(2)^3 = 10 - 3(8) = 10 - 24 = -14$. Therefore, the limit is -14.

1. Direct substitution yields 0/0. 2. Factor the numerator: $x^2 - 1 = (x - 1)(x + 1)$. 3. Simplify: $\frac{(x - 1)(x + 1)}{x + 1} = x - 1$. 4. Evaluate the limit: $lim_{x o -1} (x - 1) = -1 - 1 = -2$.