Given that limt →2 f(x)=4 limx →2 g(x)=-2 limx →2 h(x)=0...

Given that $$\lim _{t \rightarrow 2} f(x)=4 \quad \lim _{x \rightarrow 2} g(x)=-2 \quad \lim _{x \rightarrow 2} h(x)=0$$ find the limits that exist. If the limit does not exist, explain why. (a) $\lim _{t \rightarrow 2}[f(x)+5 g(x)]$ (b) $\lim _{x \rightarrow 2}[g(x)]^{3}$ (c) $\lim _{x \rightarrow 2} \sqrt{f(x)}$ (d) $\lim _{x \rightarrow 2} \frac{3 f(x)}{g(x)}$ (e) $\lim _{x \rightarrow 2} \frac{g(x)}{h(x)}$ (f) $\lim _{x \rightarrow 2} \frac{g(x) h(x)}{f(x)}$

Given that
$$\lim _{t \rightarrow 2} f(x)=4 \quad \lim _{x \rightarrow 2} g(x)=-2 \quad \lim _{x \rightarrow 2} h(x)=0$$
find the limits that exist. If the limit does not exist, explain why.
(a) $\lim _{t \rightarrow 2}[f(x)+5 g(x)]$
(b) $\lim _{x \rightarrow 2}[g(x)]^{3}$
(c) $\lim _{x \rightarrow 2} \sqrt{f(x)}$
(d) $\lim _{x \rightarrow 2} \frac{3 f(x)}{g(x)}$
(e) $\lim _{x \rightarrow 2} \frac{g(x)}{h(x)}$
(f) $\lim _{x \rightarrow 2} \frac{g(x) h(x)}{f(x)}$

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Key Concepts

Quotient Rule for Limits

The quotient rule states that if the limit of the numerator and denominator exist and the limit of the denominator is not zero, the limit of the quotient equals the quotient of the limits. This rule is central when evaluating limits of rational expressions.

Indeterminate Forms and Division by Zero

When direct substitution results in a zero in the denominator or leads to forms like 0/0 or nonzero/0, the limit might not exist or may be unbounded. Recognizing these cases is crucial for correctly interpreting the behavior of a function near a particular point.

Limit Laws

These laws state that if individual limits exist, then the limit of a sum is the sum of the limits, the limit of a constant times a function is the constant times the limit, and similarly for products and powers. They allow us to break down the analysis of more complex limit expressions into simpler parts.

Continuity

Continuity is a property of functions where the limit at a point equals the function’s value at that point. This concept justifies substituting the limit value into functions like the square root when the function is continuous, thereby simplifying limit calculations.

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