Find the slowest growing and the fastest growing functions as x→∞ .
-$$\begin{array} { l }
y = 7 x ^ { 10 } \
y = e ^ { x } \
y = e ^ { x - 2 } \
y = x e ^ { x }
\end{array}$$
A) Slowest: $y = 7 x ^ { 10 }$
Fastest: $y = e ^ { x }$ and $y = e ^ { x - 2 }$ grow at the same rate.
B) Slowest: $y = x e ^ { x }$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$
C) Slowest: $y = e ^ { x - 2 }$
Fastest: $y = x e ^ { x }$
D) Slowest: $\mathrm { y } = 7 \mathrm { x } ^ { 10 }$
Fastest: $y = x e ^ { x }$

Question

Find the slowest growing and the fastest growing functions as x→∞ .
-$$\begin{array} { l } 
y = 7 x ^ { 10 } \
y = e ^ { x } \
y = e ^ { x - 2 } \
y = x e ^ { x }
\end{array}$$
A) Slowest: $y = 7 x ^ { 10 }$
Fastest: $y = e ^ { x }$ and $y = e ^ { x - 2 }$ grow at the same rate.
B) Slowest: $y = x e ^ { x }$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$
C) Slowest: $y = e ^ { x - 2 }$
Fastest: $y = x e ^ { x }$
D) Slowest: $\mathrm { y } = 7 \mathrm { x } ^ { 10 }$
Fastest: $y = x e ^ { x }$

Quizplus · Accepted Answer

The slowest growing function is $y = 7x^{10}$ because polynomial functions grow slower than exponential functions. The fastest growing function is $y = xe^x$ because it combines both linear and exponential growth, outpacing pure exponential growth as $x \rightarrow \infty$.

Find the Slowest Growing and the Fastest Growing Functions as X→∞