Evaluating $limlimits_{x to infty}frac{4^{x+2}+3^x}{4^{x-2}}$
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$$lim_{x→∞}frac{4^{x+2}+3^x}{4^{x-2}}.$$
I have solved it like below:
$$lim_{x→∞}left(frac{4^{x+2}}{4^{x-2}}+frac{3^x}{4^{x-2}}right)=lim_{x→∞}left(4^4+frac{3^x}{4^x}·4^2right).$$
Since, as $x → ∞$, $3^x → ∞$, $dfrac{3^x}{4^x} → 0$, the limit is equal to $4^4=256$.
Have I solved it correctly?
This was a practice test question and the given solution was wrong. So, I solved it and I am preparing alone, no friend to discuss, so I posted it here.
calculus limits
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add a comment |
$begingroup$
$$lim_{x→∞}frac{4^{x+2}+3^x}{4^{x-2}}.$$
I have solved it like below:
$$lim_{x→∞}left(frac{4^{x+2}}{4^{x-2}}+frac{3^x}{4^{x-2}}right)=lim_{x→∞}left(4^4+frac{3^x}{4^x}·4^2right).$$
Since, as $x → ∞$, $3^x → ∞$, $dfrac{3^x}{4^x} → 0$, the limit is equal to $4^4=256$.
Have I solved it correctly?
This was a practice test question and the given solution was wrong. So, I solved it and I am preparing alone, no friend to discuss, so I posted it here.
calculus limits
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2
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$lim_{x to infty} left( 4^4+frac{3^x}{4^x} times 4^2 right)=lim_{x to infty}4^4 +lim_{x to infty}left(dfrac{3^x}{4^x}right)4^2$? While its true you might want to explain for rigor.
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– Yadati Kiran
Dec 22 '18 at 4:26
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It's unclear to me what purpose it serves to write $3^xtoinfty$. It makes the next expression, $frac{3^x}{4^x} to 0$, look wrong even though it actually is correct.
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– David K
Jan 8 at 14:28
add a comment |
$begingroup$
$$lim_{x→∞}frac{4^{x+2}+3^x}{4^{x-2}}.$$
I have solved it like below:
$$lim_{x→∞}left(frac{4^{x+2}}{4^{x-2}}+frac{3^x}{4^{x-2}}right)=lim_{x→∞}left(4^4+frac{3^x}{4^x}·4^2right).$$
Since, as $x → ∞$, $3^x → ∞$, $dfrac{3^x}{4^x} → 0$, the limit is equal to $4^4=256$.
Have I solved it correctly?
This was a practice test question and the given solution was wrong. So, I solved it and I am preparing alone, no friend to discuss, so I posted it here.
calculus limits
$endgroup$
$$lim_{x→∞}frac{4^{x+2}+3^x}{4^{x-2}}.$$
I have solved it like below:
$$lim_{x→∞}left(frac{4^{x+2}}{4^{x-2}}+frac{3^x}{4^{x-2}}right)=lim_{x→∞}left(4^4+frac{3^x}{4^x}·4^2right).$$
Since, as $x → ∞$, $3^x → ∞$, $dfrac{3^x}{4^x} → 0$, the limit is equal to $4^4=256$.
Have I solved it correctly?
This was a practice test question and the given solution was wrong. So, I solved it and I am preparing alone, no friend to discuss, so I posted it here.
calculus limits
calculus limits
edited Dec 22 '18 at 7:00
Saad
19.7k92352
19.7k92352
asked Dec 22 '18 at 4:14
user3767495user3767495
3888
3888
2
$begingroup$
$lim_{x to infty} left( 4^4+frac{3^x}{4^x} times 4^2 right)=lim_{x to infty}4^4 +lim_{x to infty}left(dfrac{3^x}{4^x}right)4^2$? While its true you might want to explain for rigor.
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– Yadati Kiran
Dec 22 '18 at 4:26
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It's unclear to me what purpose it serves to write $3^xtoinfty$. It makes the next expression, $frac{3^x}{4^x} to 0$, look wrong even though it actually is correct.
$endgroup$
– David K
Jan 8 at 14:28
add a comment |
2
$begingroup$
$lim_{x to infty} left( 4^4+frac{3^x}{4^x} times 4^2 right)=lim_{x to infty}4^4 +lim_{x to infty}left(dfrac{3^x}{4^x}right)4^2$? While its true you might want to explain for rigor.
$endgroup$
– Yadati Kiran
Dec 22 '18 at 4:26
$begingroup$
It's unclear to me what purpose it serves to write $3^xtoinfty$. It makes the next expression, $frac{3^x}{4^x} to 0$, look wrong even though it actually is correct.
$endgroup$
– David K
Jan 8 at 14:28
2
2
$begingroup$
$lim_{x to infty} left( 4^4+frac{3^x}{4^x} times 4^2 right)=lim_{x to infty}4^4 +lim_{x to infty}left(dfrac{3^x}{4^x}right)4^2$? While its true you might want to explain for rigor.
$endgroup$
– Yadati Kiran
Dec 22 '18 at 4:26
$begingroup$
$lim_{x to infty} left( 4^4+frac{3^x}{4^x} times 4^2 right)=lim_{x to infty}4^4 +lim_{x to infty}left(dfrac{3^x}{4^x}right)4^2$? While its true you might want to explain for rigor.
$endgroup$
– Yadati Kiran
Dec 22 '18 at 4:26
$begingroup$
It's unclear to me what purpose it serves to write $3^xtoinfty$. It makes the next expression, $frac{3^x}{4^x} to 0$, look wrong even though it actually is correct.
$endgroup$
– David K
Jan 8 at 14:28
$begingroup$
It's unclear to me what purpose it serves to write $3^xtoinfty$. It makes the next expression, $frac{3^x}{4^x} to 0$, look wrong even though it actually is correct.
$endgroup$
– David K
Jan 8 at 14:28
add a comment |
2 Answers
2
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oldest
votes
$begingroup$
Yup! Your steps almost all follow logically and evaluate to the correct limit, so your solution is correct (almost).
I will make a nitpick with one thing you said, though:
Since, as $x to infty, 3^x to infty, frac{3^x}{4^x} to 0$
Technically, since $3^x$ and $4^x$ both approach $infty$ as $x to infty$, then under your logic we obtain an indeterminate form:
$$frac{3^x}{4^x} to frac{infty}{infty}$$
It would be better to regroup $3^x/4^x$ as $(3/4)^x$. Then clearly, as $x to infty$, $(3/4)^x to 0$ because $3/4 < 1$. (If you're not convinced, notice if you take $x = 1, 2, 3, 4$ and so on that $(3/4)^x$ clearly is decreasing.)
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add a comment |
$begingroup$
Yup, the answer is correct.
Remark:
- While $3^x to infty$ is true, note that it is not used in the proof though.
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yup! Your steps almost all follow logically and evaluate to the correct limit, so your solution is correct (almost).
I will make a nitpick with one thing you said, though:
Since, as $x to infty, 3^x to infty, frac{3^x}{4^x} to 0$
Technically, since $3^x$ and $4^x$ both approach $infty$ as $x to infty$, then under your logic we obtain an indeterminate form:
$$frac{3^x}{4^x} to frac{infty}{infty}$$
It would be better to regroup $3^x/4^x$ as $(3/4)^x$. Then clearly, as $x to infty$, $(3/4)^x to 0$ because $3/4 < 1$. (If you're not convinced, notice if you take $x = 1, 2, 3, 4$ and so on that $(3/4)^x$ clearly is decreasing.)
$endgroup$
add a comment |
$begingroup$
Yup! Your steps almost all follow logically and evaluate to the correct limit, so your solution is correct (almost).
I will make a nitpick with one thing you said, though:
Since, as $x to infty, 3^x to infty, frac{3^x}{4^x} to 0$
Technically, since $3^x$ and $4^x$ both approach $infty$ as $x to infty$, then under your logic we obtain an indeterminate form:
$$frac{3^x}{4^x} to frac{infty}{infty}$$
It would be better to regroup $3^x/4^x$ as $(3/4)^x$. Then clearly, as $x to infty$, $(3/4)^x to 0$ because $3/4 < 1$. (If you're not convinced, notice if you take $x = 1, 2, 3, 4$ and so on that $(3/4)^x$ clearly is decreasing.)
$endgroup$
add a comment |
$begingroup$
Yup! Your steps almost all follow logically and evaluate to the correct limit, so your solution is correct (almost).
I will make a nitpick with one thing you said, though:
Since, as $x to infty, 3^x to infty, frac{3^x}{4^x} to 0$
Technically, since $3^x$ and $4^x$ both approach $infty$ as $x to infty$, then under your logic we obtain an indeterminate form:
$$frac{3^x}{4^x} to frac{infty}{infty}$$
It would be better to regroup $3^x/4^x$ as $(3/4)^x$. Then clearly, as $x to infty$, $(3/4)^x to 0$ because $3/4 < 1$. (If you're not convinced, notice if you take $x = 1, 2, 3, 4$ and so on that $(3/4)^x$ clearly is decreasing.)
$endgroup$
Yup! Your steps almost all follow logically and evaluate to the correct limit, so your solution is correct (almost).
I will make a nitpick with one thing you said, though:
Since, as $x to infty, 3^x to infty, frac{3^x}{4^x} to 0$
Technically, since $3^x$ and $4^x$ both approach $infty$ as $x to infty$, then under your logic we obtain an indeterminate form:
$$frac{3^x}{4^x} to frac{infty}{infty}$$
It would be better to regroup $3^x/4^x$ as $(3/4)^x$. Then clearly, as $x to infty$, $(3/4)^x to 0$ because $3/4 < 1$. (If you're not convinced, notice if you take $x = 1, 2, 3, 4$ and so on that $(3/4)^x$ clearly is decreasing.)
answered Dec 22 '18 at 4:29
Eevee TrainerEevee Trainer
6,0931936
6,0931936
add a comment |
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$begingroup$
Yup, the answer is correct.
Remark:
- While $3^x to infty$ is true, note that it is not used in the proof though.
$endgroup$
add a comment |
$begingroup$
Yup, the answer is correct.
Remark:
- While $3^x to infty$ is true, note that it is not used in the proof though.
$endgroup$
add a comment |
$begingroup$
Yup, the answer is correct.
Remark:
- While $3^x to infty$ is true, note that it is not used in the proof though.
$endgroup$
Yup, the answer is correct.
Remark:
- While $3^x to infty$ is true, note that it is not used in the proof though.
answered Dec 22 '18 at 4:26
Siong Thye GohSiong Thye Goh
101k1466118
101k1466118
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2
$begingroup$
$lim_{x to infty} left( 4^4+frac{3^x}{4^x} times 4^2 right)=lim_{x to infty}4^4 +lim_{x to infty}left(dfrac{3^x}{4^x}right)4^2$? While its true you might want to explain for rigor.
$endgroup$
– Yadati Kiran
Dec 22 '18 at 4:26
$begingroup$
It's unclear to me what purpose it serves to write $3^xtoinfty$. It makes the next expression, $frac{3^x}{4^x} to 0$, look wrong even though it actually is correct.
$endgroup$
– David K
Jan 8 at 14:28