Find $\lim _{x \rightarrow \infty} h(x)$, if it exists. (If an answer does not exist, enter DNE.)
$f(x)=9 x^3-2$
(a)
$\begin{array}{l}
h(x)=\frac{f(x)}{x^2} \\
\lim _{x \rightarrow \infty} h(x)=\square
\end{array}$
(b) $h(x)=\frac{f(x)}{x^3}$
$\lim _{x \rightarrow \infty} h(x)=\square$
(c) $h(x)=\frac{f(x)}{x^4}$
$\lim _{x \rightarrow \infty} h(x)=\square$
Answer (2)
(a) Substitute f ( x ) into h ( x ) to get h ( x ) = x 2 9 x 3 − 2 , simplify to h ( x ) = 9 x − x 2 2 , and find the limit as x approaches infinity, which is DNE.
(b) Substitute f ( x ) into h ( x ) to get h ( x ) = x 3 9 x 3 − 2 , simplify to h ( x ) = 9 − x 3 2 , and find the limit as x approaches infinity, which is 9.
(c) Substitute f ( x ) into h ( x ) to get h ( x ) = x 4 9 x 3 − 2 , simplify to h ( x ) = x 9 − x 4 2 , and find the limit as x approaches infinity, which is 0.
The limits are DNE, 9, and 0, respectively. D NE , 9 , 0
Explanation
Problem Analysis We are asked to find the limit of h ( x ) as x approaches infinity for three different definitions of h ( x ) , given that f ( x ) = 9 x 3 − 2 .
Part (a) Solution (a) Here, h ( x ) = x 2 f ( x ) = x 2 9 x 3 − 2 . We can simplify this by dividing each term in the numerator by x 2 to get h ( x ) = 9 x − x 2 2 . As x approaches infinity, 9 x also approaches infinity, and x 2 2 approaches 0. Therefore, lim x → ∞ h ( x ) = lim x → ∞ ( 9 x − x 2 2 ) = ∞ . Since the limit does not exist as a finite number, we write DNE (Does Not Exist).
Part (b) Solution (b) Here, h ( x ) = x 3 f ( x ) = x 3 9 x 3 − 2 . We can simplify this by dividing each term in the numerator by x 3 to get h ( x ) = 9 − x 3 2 . As x approaches infinity, x 3 2 approaches 0. Therefore, lim x → ∞ h ( x ) = lim x → ∞ ( 9 − x 3 2 ) = 9 − 0 = 9 .
Part (c) Solution (c) Here, h ( x ) = x 4 f ( x ) = x 4 9 x 3 − 2 . We can simplify this by dividing each term in the numerator by x 4 to get h ( x ) = x 9 − x 4 2 . As x approaches infinity, both x 9 and x 4 2 approach 0. Therefore, lim x → ∞ h ( x ) = lim x → ∞ ( x 9 − x 4 2 ) = 0 − 0 = 0 .
Final Answer Therefore, the limits are: (a) DNE (b) 9 (c) 0
Examples
Understanding limits is crucial in many real-world applications. For example, in physics, when analyzing the motion of an object, we might want to know its terminal velocity. This involves finding the limit of the object's velocity as time approaches infinity. Similarly, in engineering, limits are used to determine the stability of systems and to optimize designs. In economics, limits can help predict long-term trends and behaviors in markets. These examples show how the concept of limits, which might seem abstract, has practical and important implications in various fields.
The limits for the expressions of h ( x ) as x approaches infinity are: (a) DNE, (b) 9, and (c) 0.
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