Given that limx →a f(x)=2, limx →a g(x)=-4, limx →a h(x)=0 find the limits. (a) limx →

step 1 Problem number one, question A, limit of x tends to a of f f x plus two j of x is equal to limit

Given that limx →a f(x)=2, limx →a g(x)=-4, limx →a...

Given that $$ \lim _{x \rightarrow a} f(x)=2, \quad \lim _{x \rightarrow a} g(x)=-4, \quad \lim _{x \rightarrow a} h(x)=0 $$ find the limits. $$ \begin{array}{l}{\text { (a) } \lim _{x \rightarrow a}[f(x)+2 g(x)]} \\ {\text { (b) } \lim _{x \rightarrow a}[h(x)-3 g(x)+1]} \\ {\text { (c) } \lim _{x \rightarrow a}[f(x) g(x)] \quad \text { (d) } \lim _{x \rightarrow a}[g(x)]^{2}} \\ {\text { (e) } \lim _{x \rightarrow a} \sqrt[3]{6+f(x)} \quad \text { (f) } \lim _{x \rightarrow a} \frac{2}{g(x)}}\end{array} $$

Given that
$$
\lim _{x \rightarrow a} f(x)=2, \quad \lim _{x \rightarrow a} g(x)=-4, \quad \lim _{x \rightarrow a} h(x)=0
$$
find the limits.
$$
\begin{array}{l}{\text { (a) } \lim _{x \rightarrow a}[f(x)+2 g(x)]} \\ {\text { (b) } \lim _{x \rightarrow a}[h(x)-3 g(x)+1]} \\ {\text { (c) } \lim _{x \rightarrow a}[f(x) g(x)] \quad \text { (d) } \lim _{x \rightarrow a}[g(x)]^{2}} \\ {\text { (e) } \lim _{x \rightarrow a} \sqrt[3]{6+f(x)} \quad \text { (f) } \lim _{x \rightarrow a} \frac{2}{g(x)}}\end{array}
$$

Key Concepts

Limit of a Function

This concept describes the behavior of a function as its input approaches a specific value. It forms the foundation for calculus by providing a rigorous way to discuss the approaching value of a function without requiring the function to be evaluated exactly at that point.

Limit Laws

Limit laws are a set of rules that allow the calculation of limits for complex functions by breaking them down into simpler parts. These include the sum, difference, product, quotient, and constant multiple laws, and they justify operations like adding or multiplying limits from individual functions.

Continuity

Continuity is the property of a function that ensures the function does not have any abrupt changes or breaks at a point. When a function is continuous at a point, the limit as the input approaches that point equals the function's actual value, which permits direct substitution in limit calculations.

Operations on Limits

This involves performing algebraic manipulations on limits using the established limit laws. It includes combining limits of functions through addition, subtraction, multiplication, and division, which simplifies the evaluation process by relying on the known limits of simpler component functions.

Radical Functions and Limits

When dealing with limits that involve radical expressions, such as square roots or cube roots, the continuity of these radical functions plays an important role. Since many radical functions are continuous on their domains, limits can often be computed by directly substituting the limit values into the radical expression.

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