$y = ln x$ with their $x$ coordinates as $1,2$ and $t$ respectively [closed]
$begingroup$
Let A,B,P be the points on the curve $y = ln x$ with their $x$ coordinates as $1,2$ and $t$ respectively.
Find $$lim_{ttoinfty} (cos angle{BAP})$$
My try :
I am unable to apply limit.
algebra-precalculus
edited Dec 13 '18 at 12:51
amWhy
1
asked Dec 13 '18 at 12:19
faceface
184
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closed as unclear what you're asking by gt6989b, Jyrki Lahtonen, José Carlos Santos, onurcanbektas, jameselmore Dec 13 '18 at 17:10
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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$begingroup$
Let A,B,P be the points on the curve $y = ln x$ with their $x$ coordinates as $1,2$ and $t$ respectively.
Find $$lim_{ttoinfty} (cos angle{BAP})$$
My try :
I am unable to apply limit.
algebra-precalculus
edited Dec 13 '18 at 12:51
amWhy
1
asked Dec 13 '18 at 12:19
faceface
184
$endgroup$
closed as unclear what you're asking by gt6989b, Jyrki Lahtonen, José Carlos Santos, onurcanbektas, jameselmore Dec 13 '18 at 17:10
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
Is there a question somewhere? Some problems you are facing? Also, even though the problem is clearly related to graphs (as in: graph of a function), I don't think it's related to graph theory.
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– Matti P.
Dec 13 '18 at 12:23
$begingroup$
@MattiP. I have edited
$endgroup$
– face
Dec 13 '18 at 12:28
add a comment |
$begingroup$
Let A,B,P be the points on the curve $y = ln x$ with their $x$ coordinates as $1,2$ and $t$ respectively.
Find $$lim_{ttoinfty} (cos angle{BAP})$$
My try :
I am unable to apply limit.
algebra-precalculus
edited Dec 13 '18 at 12:51
amWhy
1
asked Dec 13 '18 at 12:19
faceface
184
$endgroup$
Let A,B,P be the points on the curve $y = ln x$ with their $x$ coordinates as $1,2$ and $t$ respectively.
Find $$lim_{ttoinfty} (cos angle{BAP})$$
My try :
I am unable to apply limit.
algebra-precalculus
algebra-precalculus
edited Dec 13 '18 at 12:51
amWhy
1
asked Dec 13 '18 at 12:19
faceface
184
edited Dec 13 '18 at 12:51
amWhy
1
asked Dec 13 '18 at 12:19
faceface
184
edited Dec 13 '18 at 12:51
amWhy
1
edited Dec 13 '18 at 12:51
amWhy
1
edited Dec 13 '18 at 12:51
amWhy
1
1
asked Dec 13 '18 at 12:19
faceface
184
asked Dec 13 '18 at 12:19
faceface
184
asked Dec 13 '18 at 12:19
faceface
184
184
closed as unclear what you're asking by gt6989b, Jyrki Lahtonen, José Carlos Santos, onurcanbektas, jameselmore Dec 13 '18 at 17:10
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by gt6989b, Jyrki Lahtonen, José Carlos Santos, onurcanbektas, jameselmore Dec 13 '18 at 17:10
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
Is there a question somewhere? Some problems you are facing? Also, even though the problem is clearly related to graphs (as in: graph of a function), I don't think it's related to graph theory.
$endgroup$
– Matti P.
Dec 13 '18 at 12:23
$begingroup$
@MattiP. I have edited
$endgroup$
– face
Dec 13 '18 at 12:28
add a comment |
1
$begingroup$
Is there a question somewhere? Some problems you are facing? Also, even though the problem is clearly related to graphs (as in: graph of a function), I don't think it's related to graph theory.
$endgroup$
– Matti P.
Dec 13 '18 at 12:23
$begingroup$
@MattiP. I have edited
$endgroup$
– face
Dec 13 '18 at 12:28
1
1
$begingroup$
Is there a question somewhere? Some problems you are facing? Also, even though the problem is clearly related to graphs (as in: graph of a function), I don't think it's related to graph theory.
$endgroup$
– Matti P.
Dec 13 '18 at 12:23
$begingroup$
Is there a question somewhere? Some problems you are facing? Also, even though the problem is clearly related to graphs (as in: graph of a function), I don't think it's related to graph theory.
$endgroup$
– Matti P.
Dec 13 '18 at 12:23
$begingroup$
@MattiP. I have edited
$endgroup$
– face
Dec 13 '18 at 12:28
$begingroup$
@MattiP. I have edited
$endgroup$
– face
Dec 13 '18 at 12:28
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It's easier to find the tangent of the angle: the slope of the line $AB$ is
$$
frac{ln2-ln1}{2-1}=ln2
$$
and the slope of the line $AP$ is
$$
frac{ln t-ln 1}{t-1}=frac{ln t}{t-1}
$$
If $f(t)$ denotes the measure of the angle $BAP$, then
$$
tan f(t)=frac{ln2-dfrac{ln t}{t-1}}{1+ln2 dfrac{ln t}{t-1}}
$$
which is essentially what you did. Therefore
$$
lim_{ttoinfty}tan f(t)=ln 2
$$
Now recall that, for an acute angle $alpha$,
$$
cosalpha=frac{1}{sqrt{1+tan^2alpha}}
$$
and apply theorems on limits.
answered Dec 13 '18 at 14:32
egregegreg
181k1485203
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It's easier to find the tangent of the angle: the slope of the line $AB$ is
$$
frac{ln2-ln1}{2-1}=ln2
$$
and the slope of the line $AP$ is
$$
frac{ln t-ln 1}{t-1}=frac{ln t}{t-1}
$$
If $f(t)$ denotes the measure of the angle $BAP$, then
$$
tan f(t)=frac{ln2-dfrac{ln t}{t-1}}{1+ln2 dfrac{ln t}{t-1}}
$$
which is essentially what you did. Therefore
$$
lim_{ttoinfty}tan f(t)=ln 2
$$
Now recall that, for an acute angle $alpha$,
$$
cosalpha=frac{1}{sqrt{1+tan^2alpha}}
$$
and apply theorems on limits.
answered Dec 13 '18 at 14:32
egregegreg
181k1485203
$endgroup$
add a comment |
$begingroup$
It's easier to find the tangent of the angle: the slope of the line $AB$ is
$$
frac{ln2-ln1}{2-1}=ln2
$$
and the slope of the line $AP$ is
$$
frac{ln t-ln 1}{t-1}=frac{ln t}{t-1}
$$
If $f(t)$ denotes the measure of the angle $BAP$, then
$$
tan f(t)=frac{ln2-dfrac{ln t}{t-1}}{1+ln2 dfrac{ln t}{t-1}}
$$
which is essentially what you did. Therefore
$$
lim_{ttoinfty}tan f(t)=ln 2
$$
Now recall that, for an acute angle $alpha$,
$$
cosalpha=frac{1}{sqrt{1+tan^2alpha}}
$$
and apply theorems on limits.
answered Dec 13 '18 at 14:32
egregegreg
181k1485203
$endgroup$
add a comment |
$begingroup$
It's easier to find the tangent of the angle: the slope of the line $AB$ is
$$
frac{ln2-ln1}{2-1}=ln2
$$
and the slope of the line $AP$ is
$$
frac{ln t-ln 1}{t-1}=frac{ln t}{t-1}
$$
If $f(t)$ denotes the measure of the angle $BAP$, then
$$
tan f(t)=frac{ln2-dfrac{ln t}{t-1}}{1+ln2 dfrac{ln t}{t-1}}
$$
which is essentially what you did. Therefore
$$
lim_{ttoinfty}tan f(t)=ln 2
$$
Now recall that, for an acute angle $alpha$,
$$
cosalpha=frac{1}{sqrt{1+tan^2alpha}}
$$
and apply theorems on limits.
answered Dec 13 '18 at 14:32
egregegreg
181k1485203
$endgroup$
It's easier to find the tangent of the angle: the slope of the line $AB$ is
$$
frac{ln2-ln1}{2-1}=ln2
$$
and the slope of the line $AP$ is
$$
frac{ln t-ln 1}{t-1}=frac{ln t}{t-1}
$$
If $f(t)$ denotes the measure of the angle $BAP$, then
$$
tan f(t)=frac{ln2-dfrac{ln t}{t-1}}{1+ln2 dfrac{ln t}{t-1}}
$$
which is essentially what you did. Therefore
$$
lim_{ttoinfty}tan f(t)=ln 2
$$
Now recall that, for an acute angle $alpha$,
$$
cosalpha=frac{1}{sqrt{1+tan^2alpha}}
$$
and apply theorems on limits.
answered Dec 13 '18 at 14:32
egregegreg
181k1485203
answered Dec 13 '18 at 14:32
egregegreg
181k1485203
answered Dec 13 '18 at 14:32
egregegreg
181k1485203
answered Dec 13 '18 at 14:32
egregegreg
181k1485203
181k1485203
add a comment |
add a comment |
1
$begingroup$
Is there a question somewhere? Some problems you are facing? Also, even though the problem is clearly related to graphs (as in: graph of a function), I don't think it's related to graph theory.
$endgroup$
– Matti P.
Dec 13 '18 at 12:23
$begingroup$
@MattiP. I have edited
$endgroup$
– face
Dec 13 '18 at 12:28