Question

Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. $$x^x;(x/2{)}^x$$

Short answer

Answer: The function $$x^x$$ grows faster than $${\left(\frac{x}{2}\right)}^x$$.

Step-by-step solution

Find the limit of the first function

To find the limit of the first function as x approaches infinity, we analyze the function $$x^x$$. Notice that as x increases, both the base and the exponent become larger, which means the function will grow faster and faster. Therefore, the limit of the function as x approaches infinity is: $$\underset{x\to \mathrm{\infty }}{lim}{x}^x=\mathrm{\infty }$$

Find the limit of the second function

To find the limit of the second function as x approaches infinity, we analyze the function $${\left(\frac{x}{2}\right)}^x$$. As x increases, both the base, now reduced to a fraction, and the exponent become larger. However, the base is half the value of x, making the rate of growth slower than the first function. The limit of the function as x approaches infinity is: $$\underset{x\to \mathrm{\infty }}{lim}{\left(\frac{x}{2}\right)}^x=\mathrm{\infty }$$ Both functions have a limit of infinity as x approaches infinity.

Determine which function grows faster or if they have comparable growth rates

To compare the growth rates, we will find the ratio of the two functions and analyze its limit as x approaches infinity: $$R(x)=\frac{x^x}{{\left(\frac{x}{2}\right)}^x}={\left(\frac{2}{1}\right)}^x=2^x$$ The limit of the ratio as x approaches infinity is: $$\underset{x\to \mathrm{\infty }}{lim}{2}^x=\mathrm{\infty }$$ Since the limit of the ratio of the two functions is infinity, it indicates that the first function grows significantly faster than the second function. In conclusion, the function $$x^x$$ grows faster than the function $${\left(\frac{x}{2}\right)}^x$$.

Key concepts

Exponential Growth

Exponential growth is a fascinating concept in mathematics. It occurs when the rate of growth of a mathematical function is directly proportional to its current size. This means the bigger something is, the faster it grows.
It's like a snowball rolling down a hill, growing larger and larger. In the exercise, we looked at functions like $x^x$ where the base $x$ itself grows over time and the exponent is the same, which means it's doubling or tripling its size rapidly as $x$ becomes very large.

• The base and the exponent increase - this means growth happens doubly fast.
• For large $x$, $x^x$ becomes incredibly large.

This kind of growth is much quicker than linear or quadratic growth (where only one element increases), making it powerful yet challenging to control. Just like with population growth or compound interest, it doubles or even triples seemingly in the blink of an eye!

Growth Rate Comparison

Comparing growth rates involves determining how quickly two functions become large relative to each other. In the exercise, we compared $x^x$ and $(x/2{)}^x$.

• To decide which grows faster, we compared their limits as $x$ went to infinity.
• Instead of just their individual limits, comparing their ratio $R(x)=\frac{x^x}{(x/2{)}^x}$ helps.
• When the ratio $R(x)$ simplifies to $2^x$ and its limit is $\mathrm{\infty }$, it means $x^x$ "outraces" $(x/2{)}^x$.

Growth rate comparisons are all about seeing which function increases its value faster as $x$ becomes large. They are crucial in analyzing algorithms and processes efficiency or just understanding how phenomena scale with size.

Infinity Limits

Infinity is not a number. Instead, it's a concept that describes something without bound or end. In limits, when we say a limit approaches infinity, it implies the function keeps growing larger and larger without stop.
For the functions $\underset{x\to \mathrm{\infty }}{lim}{x}^x=\mathrm{\infty }$ and $\underset{x\to \mathrm{\infty }}{lim}(x/2{)}^x=\mathrm{\infty }$, we understand both grow infinitely as $x$ increases. However, infinity alone doesn't tell if one might get there faster than another.

• Each function's behavior when indeterminate, like $\frac{\mathrm{\infty }}{\mathrm{\infty }}$, can suggest convergence to a particular form.
• Though both tend towards infinity, the rate is what determines more potential for scaling in real-world scenarios.

So, infinity in limits helps identify unbounded growth, but the accompanying rate provides deeper insights into how functions compare over large domains of $x$. It's essential in physics and many sciences to understand a system's potential long-term behavior.