Find the slowest growing and the fastest growing functions as x→∞ .
-$$\begin{array} { l }
y = 2 x ^ { 2 } + 9 x \
y = e ^ { x } \
y = e ^ { x } / 6 \
y = \log _ { 4 } x
\end{array}$$
A) Slowest: $y = \log _ { 4 } x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { X } }$
B) Slowest: $2 x ^ { 2 } + 9 x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$
C) Slowest: $y = e ^ { x } / 6$
Fastest: $2 x ^ { 2 } + 9 x$
D) Slowest: $y = \log _ { 4 } x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$ and $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } } / 6$ grow at the same rate.

Question

Find the slowest growing and the fastest growing functions as x→∞ .
-$$\begin{array} { l } 
y = 2 x ^ { 2 } + 9 x \
y = e ^ { x } \
y = e ^ { x } / 6 \
y = \log _ { 4 } x
\end{array}$$
A) Slowest: $y = \log _ { 4 } x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { X } }$
B) Slowest: $2 x ^ { 2 } + 9 x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$
C) Slowest: $y = e ^ { x } / 6$
Fastest: $2 x ^ { 2 } + 9 x$
D) Slowest: $y = \log _ { 4 } x$
Fastest: $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }$ and $\mathrm { y } = \mathrm { e } ^ { \mathrm { x } } / 6$ grow at the same rate.

Quizplus · Accepted Answer

The slowest growing function is $y = \log_4 x$ because logarithmic growth is slower than polynomial, which in turn is slower than exponential growth. The fastest growing functions are $y = e^x$ and $y = e^x / 6$, both of which are exponential and grow at the same rate since dividing by a constant does not change the growth rate.

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