Let g(t)={ (2+3 t)/(5 t^2+6), t 1,000,000 . Find

step 1 Okay, so this time we're given G of T, well, this here is a piecewise function. And for part A,

Let g(t)={ (2+3 t)/(5 t^2+6), t<1,000,000 (√(36...

Let $$ g(t)=\left\{\begin{array}{ll}{\frac{2+3 t}{5 t^{2}+6},} & {t<1,000,000} \\ {\frac{\sqrt{36 t^{2}-100}}{5-t},} & {t>1,000,000}\end{array}\right. $$ Find $$ \begin{array}{ll}{\text { (a) } \lim _{t \rightarrow-\infty} g(t)} & {\text { (b) } \lim _{t \rightarrow+\infty} g(t)}\end{array} $$

Let
$$
g(t)=\left\{\begin{array}{ll}{\frac{2+3 t}{5 t^{2}+6},} & {t<1,000,000} \\ {\frac{\sqrt{36 t^{2}-100}}{5-t},} & {t>1,000,000}\end{array}\right.
$$
Find
$$
\begin{array}{ll}{\text { (a) } \lim _{t \rightarrow-\infty} g(t)} & {\text { (b) } \lim _{t \rightarrow+\infty} g(t)}\end{array}
$$

Key Concepts

Limit at Infinity

This concept involves analyzing the behavior of a function as the input variable grows without bound (either positively or negatively). When finding limits at infinity, the focus is typically on understanding the dominant terms in the numerator and denominator that influence the function's behavior as the variable becomes very large, leading to conclusions about horizontal asymptotes or unbounded behavior.

Dominant Term Analysis

In the context of limits at infinity, dominant term analysis is a technique where we identify the terms in an expression that grow the fastest as the variable approaches infinity. These terms determine the behavior of the function, while lower-order terms become negligible. This method is particularly useful when evaluating rational functions or expressions involving square roots of polynomials.

Rational Functions

Rational functions are functions expressed as the ratio of two polynomials. The behavior of these functions as the variable approaches infinity or negative infinity is often determined by comparing the highest power terms in the numerator and the denominator. This comparison is key to finding horizontal asymptotes and evaluating limits at infinity.

Piecewise Functions

Piecewise functions are defined by different expressions over different parts of their domain. When calculating limits for such functions, it is important to consider the specific expression applicable to the interval of interest. In this context, calculating limits as the variable approaches positive or negative infinity requires using the appropriate piece of the function that governs the behavior in that region.

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