1. Which of the following functions grows the fastest as x goes to infinity?
- 2^x
- 3^x
- e^x <-- my answer
- x^20
2. Compare the rates of growth of f(x) = x + sinx and g(x) = x as x approaches infinity.
- f(x) grows faster than g(x) as x goes to infinity.
- g(x) grows faster than f(x) as x goes to infinity.
- f(x) and g(x) grow at the same rate as x goes to infinity.
- The rate of growth cannot be determined.
3. What does lim x --> inf f(x)/g(x) = 5 show?
- g(x) grows faster than f(x) as x goes to infinity.
- f(x) and g(x) grow at the same rate as x goes to infinity. <-- my answer
- f(x) grows faster than g(x) as x goes to infinity.
- L'Hôpital's Rule must be used to determine the true limit value.
4. Which of the following functions grows at the same rate as 3x as x goes to infinity?
- 2x
- √3^x+4 <-- my answer
- √6^x
- √9^x+5
5. Which of the following functions grows the slowest as x goes to infinity?
- 5^x <-- my answer
- 5^x
- x^5
- They all grow at the same rate.
Oh, so you want to talk about functions, huh? Well, buckle up, because I'm about to take you on a wild ride through the wacky world of growth rates!
1. Which function grows the fastest as x goes to infinity? Well, let's see. We have 2^x, 3^x, e^x, and x^20. Now, if we had a race between these functions, it would go something like this: 2^x would be running along, thinking it's pretty fast, when suddenly 3^x comes zooming past, leaving it in the dust. But just when 3^x is celebrating its victory, along comes e^x, soaring through the sky like a superhero on steroids. And poor x^20, well, it's still trying to tie its shoelaces. So, my friend, the winner of this race is indeed e^x! It grows the fastest as x goes to infinity.
2. Ah, the good old comparison of growth rates. We have f(x) = x + sinx and g(x) = x. Now, if f(x) and g(x) were in a race, it would be a bit like watching a tortoise and a hare race. You see, f(x) is like the tortoise, steadily plodding along, occasionally getting a boost from its buddy sinx. Meanwhile, g(x) is like the hare, just cruising along at a constant speed, not really caring about sinusoidal shenanigans. Now, if we imagine them racing to infinity, it's clear that f(x) will eventually surpass g(x) because of sinx's occasional boosts. So, my friend, f(x) grows faster than g(x) as x goes to infinity.
3. Ah, limits, the bane of every calculus student's existence. Now, we have lim x --> inf f(x)/g(x) = 5. This means that as x goes to infinity, the ratio of f(x) to g(x) approaches 5. But what does that tell us about their growth rates? Well, think of it like this: f(x) and g(x) are like two people running a race. If their ratio approaches 5, it means that, on average, f(x) is growing at roughly the same rate as g(x). They might have their ups and downs, but when it's all said and done, they both cross the finish line at around the same time. So, my friend, f(x) and g(x) grow at the same rate as x goes to infinity.
4. Ah, the joys of finding functions that grow at the same rate as our good old friend 3x. We have some contenders here: 2x, √3^x+4, √6^x, and √9^x+5. Now, if we're looking for a function that grows at the same rate as 3x, we need something that keeps up with the constant factor of 3. And if we take a close look, we see that the square root of 3^x+4 does just that. It's like a ninja, sneaking in and matching 3x step for step. So, my friend, the function √3^x+4 grows at the same rate as 3x as x goes to infinity.
5. Ah, the eternal question of who grows the slowest. We have 5^x, 5^x, x^5, and they all grow at the same rate. Well, my friend, let me break it to you gently: 5^x grows faster than a speeding bullet. It's like the Flash on steroids, zooming past anything that dares to stand in its way. Meanwhile, poor x^5 is just chugging along, trying its best but ultimately falling behind. So, my friend, the slowest of the bunch is x^5. It's like the sloth of the math world, taking its sweet, sweet time to grow.
Well, I hope I've brought some humor and clarity to this journey through growth rates. Remember, in the wacky world of math, there's always a surprise waiting around the corner. Keep embracing the madness, my friend!
1. The correct answer is "3^x". As x goes to infinity, exponential functions with larger bases grow faster. Therefore, 3^x grows faster than 2^x, e^x, and x^20.
2. The correct answer is "f(x) grows faster than g(x) as x goes to infinity." When evaluating the growth rate, we consider the dominant term. In this case, both f(x) and g(x) have a dominant term of x. However, the presence of the sin(x) term in f(x) introduces oscillations, resulting in a faster growth rate compared to g(x) which is a linear function.
3. The correct answer is "f(x) and g(x) grow at the same rate as x goes to infinity." The limit expression f(x)/g(x) = 5 indicates that the ratio of f(x) to g(x) approaches a constant value of 5 as x goes to infinity. This means that both f(x) and g(x) have the same growth rate in this limit.
4. The correct answer is "√3^x+4". To identify a function that grows at the same rate as 3x, we need to compare the exponent terms since the base 3 remains the same. Among the given options, √3^x+4 has a square root that cancels the exponent term x+4, resulting in a growth rate similar to 3x.
5. The correct answer is "They all grow at the same rate." In this case, all the options have an exponential term with the base 5. Exponential functions with the same base grow at the same rate as x goes to infinity. Therefore, all options grow at the same rate.
1. To determine which function grows the fastest, we can compare their growth rates by evaluating the limit as x approaches infinity for each function. Let's evaluate the limit for each option:
- 2^x: As x approaches infinity, 2^x grows exponentially, getting larger and larger.
- 3^x: Similar to 2^x, as x approaches infinity, 3^x also grows exponentially.
- e^x: The exponential function e^x also grows exponentially, but it has a constant base value of e (approximately 2.718), which is greater than 2 and 3.
- x^20: The polynomial function x^20 grows, but at a slower rate compared to the exponential functions above.
Therefore, the correct answer is e^x, as it grows faster than the other options.
2. To compare the growth rates of f(x) = x + sinx and g(x) = x as x approaches infinity, we can again evaluate the limit as x goes to infinity for both functions.
- As x approaches infinity, the sine function oscillates between -1 and 1. Thus, sinx has a bounded behavior.
- On the other hand, x grows linearly as x approaches infinity.
Based on these observations, we can conclude that g(x) = x grows faster than f(x) = x + sinx as x approaches infinity.
3. The limit lim x --> inf f(x)/g(x) = 5 suggests that as x approaches infinity, f(x) and g(x) grow at the same rate. Since the limit equals a finite value (5 in this case), it implies that the two functions have a matching growth pattern as x becomes large. Therefore, the correct answer is that f(x) and g(x) grow at the same rate as x goes to infinity.
4. To identify which function grows at the same rate as 3x as x goes to infinity, we need to evaluate the limit as x approaches infinity for each option.
- 2x: This function grows at a slower rate than 3x, so it is not the correct answer.
- √3^x+4: This function involves raising 3 to a power, which grows at a faster rate than 3x.
- √6^x: Similar to the previous option, this function grows faster than 3x.
- √9^x+5: Again, this function also grows faster than 3x.
Therefore, the correct answer is √3^x+4, as it grows at the same rate as 3x as x goes to infinity.
5. To determine which function grows the slowest as x goes to infinity, we can again compare their growth rates by evaluating the limit as x approaches infinity for each function. Let's evaluate the limit for each option:
- 5^x: As x approaches infinity, 5^x grows exponentially, getting larger and larger.
- x^5: The polynomial function x^5 grows, but at a slower rate compared to exponential growth.
Based on these observations, we can conclude that x^5 grows slower than 5^x. Therefore, the correct answer is x^5, as it grows slower than the other options.
see
https://www.jiskha.com/display.cgi?id=1530619883