Find each limit, if possible. (a) limx →∞ (5 x^3 / 2)/(4 x^2+1) (b) limx →∞ (5 x^3 / 2)/(4 x^3 /

step 1 Starting with part A, we're finding the limit as X goes to infinity of the quantity of 5x to the

Find each limit, if possible. (a) limx →∞ (5 x^3 / 2)/(4...

Find each limit, if possible.
(a) $\lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 x^{2}+1}$
(b) $\lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 x^{3 / 2}+1}$
(c) $\lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 \sqrt{x}+1}$

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Key Concepts

Limits at Infinity

This concept involves evaluating the behavior of functions as the independent variable grows without bound. In such cases, the focus is on identifying which terms contribute the most to the function's value at large inputs, allowing one to determine the limit of the function.

Dominant Terms

Dominant terms are those parts of an expression, typically those with the highest exponent in polynomial functions, that dictate the behavior of the expression as the variable tends to infinity. In rational functions, comparing the dominant terms in the numerator and denominator is essential for determining the limit.

Growth Rates of Functions

The growth rate of a function refers to how quickly its value increases as the input becomes large, which is often determined by its exponent in polynomial or power functions. When comparing two functions in a limit, if one grows faster than the other, it can lead to the limit being zero or infinity, or equivalently finite if the rates are the same.

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