Find the slowest growing and the fastest growing functions as x→∞ .
-$$\begin{array} { l }
y = \ln 2 x \
y = 5 \ln x \
y = \frac { 1 } { x } \
y = \sqrt { x }
\end{array}$$
A) Slowest: $y = \frac { 1 } { x }$
Fastest: $y = \sqrt { x }$
B) Slowest: $y = \frac { 1 } { x }$
Fastest: $y = 5 \ln x$
C) Slowest: $y = \sqrt { x }$
Fastest: $y = \ln 2 x$ and $y = 5 \ln x$ grow at the same rate.
D) Slowest: $y = \ln 2 x$ and $y = 5 \ln x$ grow at the same rate. Fastest: $y = \sqrt { x }$

Question

Find the slowest growing and the fastest growing functions as x→∞ .
-$$\begin{array} { l } 
y = \ln 2 x \
y = 5 \ln x \
y = \frac { 1 } { x } \
y = \sqrt { x }
\end{array}$$
A) Slowest: $y = \frac { 1 } { x }$
Fastest: $y = \sqrt { x }$
B) Slowest: $y = \frac { 1 } { x }$
Fastest: $y = 5 \ln x$
C) Slowest: $y = \sqrt { x }$
Fastest: $y = \ln 2 x$ and $y = 5 \ln x$ grow at the same rate.
D) Slowest: $y = \ln 2 x$ and $y = 5 \ln x$ grow at the same rate. Fastest: $y = \sqrt { x }$

Quizplus · Accepted Answer

As x approaches infinity, the slowest growing function is always any function of the form $\frac { 1 } { x^k }$ where k is a positive constant. This means that $y = \frac { 1 } { x }$ is the slowest. On the other hand, the fastest-growing function is always any function of the form $x^k$ where k is a positive constant. This means that $y = \sqrt { x }$ is the fastest.

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