Does L'Hopital's Rule extend to $x rightarrow infty$ and $L= infty?$
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The following is given:
Suppose that $f$ and $g$ are continuous functions on the interval $(a,b)$ and that $$lim_{x rightarrow b} frac{f(x)}{g(x)} = frac{0}{0}.$$ If $g(x),g'(x) neq 0$ for all $x in (a,b)$ and $$lim_{x rightarrow b} frac{f'(x)}{g'(x)} = L.$$ Then $$lim_{x rightarrow b} frac{f(x)}{g(x)} = L.$$ It is also given that if $x rightarrow infty$ (rather than $x rightarrow b$), the property holds.
Does the property hold if $x rightarrow infty$ and $L = infty?$
calculus real-analysis limits
asked Nov 13 at 18:46
Rafael Vergnaud
296116
add a comment |
up vote
0
down vote
favorite
The following is given:
Suppose that $f$ and $g$ are continuous functions on the interval $(a,b)$ and that $$lim_{x rightarrow b} frac{f(x)}{g(x)} = frac{0}{0}.$$ If $g(x),g'(x) neq 0$ for all $x in (a,b)$ and $$lim_{x rightarrow b} frac{f'(x)}{g'(x)} = L.$$ Then $$lim_{x rightarrow b} frac{f(x)}{g(x)} = L.$$ It is also given that if $x rightarrow infty$ (rather than $x rightarrow b$), the property holds.
Does the property hold if $x rightarrow infty$ and $L = infty?$
calculus real-analysis limits
asked Nov 13 at 18:46
Rafael Vergnaud
296116
1
It's best to not write things like $$lim_{x rightarrow b} frac{f(x)}{g(x)} = frac{0}{0}.$$ (Also, you want $xto b^-.$)
– zhw.
Nov 13 at 18:57
Well, the version where $L=infty$ is true and very crucial in some circumstances.
– Paramanand Singh
Nov 13 at 19:02
How would you write it?
– Rafael Vergnaud
Nov 13 at 19:17
Just write $lim_{xto b-} f(x) = 0$ and $lim_{xto b-} g(x)=0$.
– Richard Martin
Nov 14 at 9:06
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The following is given:
Suppose that $f$ and $g$ are continuous functions on the interval $(a,b)$ and that $$lim_{x rightarrow b} frac{f(x)}{g(x)} = frac{0}{0}.$$ If $g(x),g'(x) neq 0$ for all $x in (a,b)$ and $$lim_{x rightarrow b} frac{f'(x)}{g'(x)} = L.$$ Then $$lim_{x rightarrow b} frac{f(x)}{g(x)} = L.$$ It is also given that if $x rightarrow infty$ (rather than $x rightarrow b$), the property holds.
Does the property hold if $x rightarrow infty$ and $L = infty?$
calculus real-analysis limits
asked Nov 13 at 18:46
Rafael Vergnaud
296116
The following is given:
Suppose that $f$ and $g$ are continuous functions on the interval $(a,b)$ and that $$lim_{x rightarrow b} frac{f(x)}{g(x)} = frac{0}{0}.$$ If $g(x),g'(x) neq 0$ for all $x in (a,b)$ and $$lim_{x rightarrow b} frac{f'(x)}{g'(x)} = L.$$ Then $$lim_{x rightarrow b} frac{f(x)}{g(x)} = L.$$ It is also given that if $x rightarrow infty$ (rather than $x rightarrow b$), the property holds.
Does the property hold if $x rightarrow infty$ and $L = infty?$
calculus real-analysis limits
calculus real-analysis limits
asked Nov 13 at 18:46
Rafael Vergnaud
296116
asked Nov 13 at 18:46
Rafael Vergnaud
296116
asked Nov 13 at 18:46
Rafael Vergnaud
296116
asked Nov 13 at 18:46
Rafael Vergnaud
296116
asked Nov 13 at 18:46
Rafael Vergnaud
296116
296116
1
It's best to not write things like $$lim_{x rightarrow b} frac{f(x)}{g(x)} = frac{0}{0}.$$ (Also, you want $xto b^-.$)
– zhw.
Nov 13 at 18:57
Well, the version where $L=infty$ is true and very crucial in some circumstances.
– Paramanand Singh
Nov 13 at 19:02
How would you write it?
– Rafael Vergnaud
Nov 13 at 19:17
Just write $lim_{xto b-} f(x) = 0$ and $lim_{xto b-} g(x)=0$.
– Richard Martin
Nov 14 at 9:06
add a comment |
1
It's best to not write things like $$lim_{x rightarrow b} frac{f(x)}{g(x)} = frac{0}{0}.$$ (Also, you want $xto b^-.$)
– zhw.
Nov 13 at 18:57
Well, the version where $L=infty$ is true and very crucial in some circumstances.
– Paramanand Singh
Nov 13 at 19:02
How would you write it?
– Rafael Vergnaud
Nov 13 at 19:17
Just write $lim_{xto b-} f(x) = 0$ and $lim_{xto b-} g(x)=0$.
– Richard Martin
Nov 14 at 9:06
1
1
It's best to not write things like $$lim_{x rightarrow b} frac{f(x)}{g(x)} = frac{0}{0}.$$ (Also, you want $xto b^-.$)
– zhw.
Nov 13 at 18:57
It's best to not write things like $$lim_{x rightarrow b} frac{f(x)}{g(x)} = frac{0}{0}.$$ (Also, you want $xto b^-.$)
– zhw.
Nov 13 at 18:57
Well, the version where $L=infty$ is true and very crucial in some circumstances.
– Paramanand Singh
Nov 13 at 19:02
Well, the version where $L=infty$ is true and very crucial in some circumstances.
– Paramanand Singh
Nov 13 at 19:02
How would you write it?
– Rafael Vergnaud
Nov 13 at 19:17
How would you write it?
– Rafael Vergnaud
Nov 13 at 19:17
Just write $lim_{xto b-} f(x) = 0$ and $lim_{xto b-} g(x)=0$.
– Richard Martin
Nov 14 at 9:06
Just write $lim_{xto b-} f(x) = 0$ and $lim_{xto b-} g(x)=0$.
– Richard Martin
Nov 14 at 9:06
add a comment |
1 Answer
1
active
oldest
votes
up vote
-1
down vote
Yes, change variable $xmapsto 1/x$ and look around $0$.
answered Nov 13 at 18:49
Richard Martin
1,4618
That was my proof for when the limit approached some finite L. How about when the limit diverges to infinity?
– Rafael Vergnaud
Nov 13 at 19:16
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
-1
down vote
Yes, change variable $xmapsto 1/x$ and look around $0$.
answered Nov 13 at 18:49
Richard Martin
1,4618
That was my proof for when the limit approached some finite L. How about when the limit diverges to infinity?
– Rafael Vergnaud
Nov 13 at 19:16
add a comment |
up vote
-1
down vote
Yes, change variable $xmapsto 1/x$ and look around $0$.
answered Nov 13 at 18:49
Richard Martin
1,4618
That was my proof for when the limit approached some finite L. How about when the limit diverges to infinity?
– Rafael Vergnaud
Nov 13 at 19:16
add a comment |
up vote
-1
down vote
up vote
-1
down vote
Yes, change variable $xmapsto 1/x$ and look around $0$.
answered Nov 13 at 18:49
Richard Martin
1,4618
Yes, change variable $xmapsto 1/x$ and look around $0$.
answered Nov 13 at 18:49
Richard Martin
1,4618
answered Nov 13 at 18:49
Richard Martin
1,4618
answered Nov 13 at 18:49
Richard Martin
1,4618
answered Nov 13 at 18:49
Richard Martin
1,4618
1,4618
That was my proof for when the limit approached some finite L. How about when the limit diverges to infinity?
– Rafael Vergnaud
Nov 13 at 19:16
add a comment |
That was my proof for when the limit approached some finite L. How about when the limit diverges to infinity?
– Rafael Vergnaud
Nov 13 at 19:16
That was my proof for when the limit approached some finite L. How about when the limit diverges to infinity?
– Rafael Vergnaud
Nov 13 at 19:16
That was my proof for when the limit approached some finite L. How about when the limit diverges to infinity?
– Rafael Vergnaud
Nov 13 at 19:16
add a comment |
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1
It's best to not write things like $$lim_{x rightarrow b} frac{f(x)}{g(x)} = frac{0}{0}.$$ (Also, you want $xto b^-.$)
– zhw.
Nov 13 at 18:57
Well, the version where $L=infty$ is true and very crucial in some circumstances.
– Paramanand Singh
Nov 13 at 19:02
How would you write it?
– Rafael Vergnaud
Nov 13 at 19:17
Just write $lim_{xto b-} f(x) = 0$ and $lim_{xto b-} g(x)=0$.
– Richard Martin
Nov 14 at 9:06