Determine the following limits at infinity.

lim t→∞ (−12t^−5)

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Identify the expression: $-12t^{-5}$.

Recognize that as $t$ approaches infinity, the term $t^{-5}$ represents $\frac{1}{t^5}$.

Understand that $\frac{1}{t^5}$ approaches 0 as $t$ approaches infinity because the denominator grows very large.

Multiply the limit of $t^{-5}$ by $-12$, which is a constant, to find the limit of the entire expression.

Conclude that the limit of $-12t^{-5}$ as $t$ approaches infinity is 0.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits at Infinity

Limits at infinity refer to the behavior of a function as the input approaches infinity. This concept is crucial in calculus for understanding how functions behave in extreme cases, particularly for rational functions, polynomials, and exponential functions. Evaluating limits at infinity helps determine horizontal asymptotes and the end behavior of functions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For example, t^−5 is equivalent to 1/t^5. Understanding negative exponents is essential for simplifying expressions and evaluating limits, especially as the variable approaches infinity, where terms with negative exponents tend to zero.

Behavior of Functions as t Approaches Infinity

The behavior of functions as t approaches infinity involves analyzing how the function's value changes as t increases without bound. In the case of polynomial functions with negative exponents, the function's value will approach zero. This concept is vital for determining the limit of functions at infinity and understanding their long-term trends.

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Textbook Question

Determine $$\lim$_{x$\rightarrow\]\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).