1-4 Given that limx → a f(x)=0 limx → a g(x)=0 limx → a h(x)=1 limx → a p(x)=∞ lim

step 1 So for the following problem, we want to determine which ones are in the indeterminate form. Fir

1-4 Given that limx → a f(x)=0 limx → a g(x)=0 limx...

1-4 Given that \begin{array}{c}{\lim _{x \rightarrow a} f(x)=0 \quad \lim _{x \rightarrow a} g(x)=0 \quad \lim _{x \rightarrow a} h(x)=1} \\ {\lim _{x \rightarrow a} p(x)=\infty \quad \lim _{x \rightarrow a} q(x)=\infty} \\ {\text { which of the following limits are indeterminate forms? For those }} \\ {\text { that are not an indeterminate form, evaluate the limit where }} \\ {\text { possible. }}\end{array} $$\quad(a)\lim _{x \rightarrow a}[f(x)]^{g(x)} \quad(b) \lim _{x \rightarrow a}[f(x)]^{p(x)} \quad(c) \lim _{x \rightarrow a}[h(x)]^{p(x)}$$ $$\quad(d) \lim _{x \rightarrow a}[p(x)]^{f(x)} \quad(e) \lim _{x \rightarrow a}[p(x)]^{q(x)} \quad(f) \lim _{x \rightarrow a} \sqrt[q(x)]{p(x)}$$

1-4 Given that
\begin{array}{c}{\lim _{x \rightarrow a} f(x)=0 \quad \lim _{x \rightarrow a} g(x)=0 \quad \lim _{x \rightarrow a} h(x)=1} \\ {\lim _{x \rightarrow a} p(x)=\infty \quad \lim _{x \rightarrow a} q(x)=\infty} \\ {\text { which of the following limits are indeterminate forms? For those }} \\ {\text { that are not an indeterminate form, evaluate the limit where }} \\ {\text { possible. }}\end{array}
$$\quad(a)\lim _{x \rightarrow a}[f(x)]^{g(x)} \quad(b) \lim _{x \rightarrow a}[f(x)]^{p(x)} \quad(c) \lim _{x \rightarrow a}[h(x)]^{p(x)}$$
$$\quad(d) \lim _{x \rightarrow a}[p(x)]^{f(x)} \quad(e) \lim _{x \rightarrow a}[p(x)]^{q(x)} \quad(f) \lim _{x \rightarrow a} \sqrt[q(x)]{p(x)}$$

Key Concepts

Behavior of Functions Near Limit Points

Understanding how functions behave as they approach certain critical values—such as zero, one, or infinity—is crucial in limit evaluation. This includes knowing the effects of limits where a function tends to zero, leading to a vanishing base, or where a function tends to infinity, leading to an unbounded influence in the exponent. The combined behavior of these functions may result in an indeterminate form that necessitates further algebraic manipulation for proper evaluation.

Indeterminate Forms

Indeterminate forms occur in limit problems when the limit expression leads to an ambiguous situation that does not allow a direct evaluation. Common examples include cases such as 0^0, 0^?, ?^0, and 1^?, where the interplay between the base and the exponent creates uncertainty about the outcome. Recognizing these forms is essential because standard limit laws do not apply directly, and one must use additional techniques to resolve the ambiguity.

Exponential Limits and Logarithmic Transformation

Many limit expressions involving exponentiation can be analyzed by taking the logarithm of the expression and then applying limit laws. This transformation converts the power form into a product of a function and the logarithm of another function. The technique is especially powerful in handling indeterminate exponential forms by reexpressing them as limits of the form g(x) log(f(x)), which can sometimes be evaluated using methods like L'Hôpital's Rule.

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