Given that limx →c f(x)=3, limx →c g(x)=0, limx →c...

Given that $\lim _{x \rightarrow c} f(x)=3, \quad \lim _{x \rightarrow c} g(x)=0, \quad \lim _{x \rightarrow c} h(x)=-2$ evaluate the limits that exist. If the limit does not exist, state how you know that. (a) $\lim _{x \rightarrow c}[3 f(x)-2 h(x)]$ (b) $\lim _{x \rightarrow c}[h(x)]^{3}$ (c) $\lim _{x \rightarrow c} \frac{h(x)}{x-c}$ (d) $\lim _{x \rightarrow c} \frac{g(x)}{h(x)}$ (e) $\lim _{x \rightarrow c} \frac{4}{f(x)-h(x)}$ (1) $\lim _{x \rightarrow \epsilon}[3+g(x)]^{2}$

Given that $\lim _{x \rightarrow c} f(x)=3, \quad \lim _{x \rightarrow c} g(x)=0, \quad \lim _{x \rightarrow c} h(x)=-2$ evaluate the limits that exist. If the limit does not exist, state how you know that.
(a) $\lim _{x \rightarrow c}[3 f(x)-2 h(x)]$
(b) $\lim _{x \rightarrow c}[h(x)]^{3}$
(c) $\lim _{x \rightarrow c} \frac{h(x)}{x-c}$
(d) $\lim _{x \rightarrow c} \frac{g(x)}{h(x)}$
(e) $\lim _{x \rightarrow c} \frac{4}{f(x)-h(x)}$
(1) $\lim _{x \rightarrow \epsilon}[3+g(x)]^{2}$

Key Concepts

Division by Zero and Undefined Limits

When a limit involves division by an expression that approaches zero while the numerator approaches a non-zero value, the limit may not exist or require additional analysis to determine if it diverges to infinity. Recognizing such scenarios is crucial in ensuring the proper evaluation of limits involving rational functions.

Direct Substitution

When individual function limits are known and the operations performed on these functions are continuous, the limit of the combined expression can be computed by directly substituting the limit values into the expression. This is a common technique in solving limit problems involving algebraic operations.

Limit Laws

These rules allow the evaluation of limits for sums, differences, constant multiples, products, quotients, and powers of functions provided that the individual limits exist and that operations such as division do not involve a zero denominator. They are fundamental tools for breaking down complex limit expressions into simpler parts that can be evaluated directly.

Continuity

Continuity of a function means that the limit of the function as it approaches a point is equal to the function's value at that point. This concept underpins the application of direct substitution in limit problems when the functions involved are known to be continuous over the interval of interest.

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