Is it sufficient to just “plug in numbers” to prove that a limit exists for multivariable functions?
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I have just learned that if $lim limits_{(x,y) to (a,b)}f(x,y)=f(a,b)$ then the function is continuous at that point.
My question is, let's say we have a very simple function and limit, such as:
$lim limits_{(x,y) to (2,3)}x^2+y^2=L$
Can I, without any further justification, just "plug in" (2,3) to get L=13? Or would I need some further proof (like the squeeze theorem). Similarly, if I can just "plug in" x and y values into a function and get a real number back, is that sufficient to conclude that the function is continuous at that point?
Also please avoid using epsilon delta stuff, I haven't learned that at all.
limits multivariable-calculus continuity
asked Nov 17 at 23:59
vevvvio
1112
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up vote
0
down vote
favorite
I have just learned that if $lim limits_{(x,y) to (a,b)}f(x,y)=f(a,b)$ then the function is continuous at that point.
My question is, let's say we have a very simple function and limit, such as:
$lim limits_{(x,y) to (2,3)}x^2+y^2=L$
Can I, without any further justification, just "plug in" (2,3) to get L=13? Or would I need some further proof (like the squeeze theorem). Similarly, if I can just "plug in" x and y values into a function and get a real number back, is that sufficient to conclude that the function is continuous at that point?
Also please avoid using epsilon delta stuff, I haven't learned that at all.
limits multivariable-calculus continuity
asked Nov 17 at 23:59
vevvvio
1112
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have just learned that if $lim limits_{(x,y) to (a,b)}f(x,y)=f(a,b)$ then the function is continuous at that point.
My question is, let's say we have a very simple function and limit, such as:
$lim limits_{(x,y) to (2,3)}x^2+y^2=L$
Can I, without any further justification, just "plug in" (2,3) to get L=13? Or would I need some further proof (like the squeeze theorem). Similarly, if I can just "plug in" x and y values into a function and get a real number back, is that sufficient to conclude that the function is continuous at that point?
Also please avoid using epsilon delta stuff, I haven't learned that at all.
limits multivariable-calculus continuity
asked Nov 17 at 23:59
vevvvio
1112
I have just learned that if $lim limits_{(x,y) to (a,b)}f(x,y)=f(a,b)$ then the function is continuous at that point.
My question is, let's say we have a very simple function and limit, such as:
$lim limits_{(x,y) to (2,3)}x^2+y^2=L$
Can I, without any further justification, just "plug in" (2,3) to get L=13? Or would I need some further proof (like the squeeze theorem). Similarly, if I can just "plug in" x and y values into a function and get a real number back, is that sufficient to conclude that the function is continuous at that point?
Also please avoid using epsilon delta stuff, I haven't learned that at all.
limits multivariable-calculus continuity
limits multivariable-calculus continuity
asked Nov 17 at 23:59
vevvvio
1112
asked Nov 17 at 23:59
vevvvio
1112
asked Nov 17 at 23:59
vevvvio
1112
asked Nov 17 at 23:59
vevvvio
1112
asked Nov 17 at 23:59
vevvvio
1112
1112
add a comment |
add a comment |
1 Answer
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No. By "plugging in numbers," what you are assuming is continuity at the point you're plugging in. If you have already shown your function is continuous (or you are allowed to assume this), then you're fine, otherwise, you need to actually show that the limit equals the value you get by plugging in those numbers. The way one traditionally does this is by "epsilon-delta stuff."
answered Nov 18 at 0:14
Carl Schildkraut
10.6k11438
Right, but I had a quiz question that asked if a particular point was continuous, and this particular point gave a real number for its output in the function. I guess my struggle here is that to show a point is continuous, I must evaluate the limit of that point. How am I supposed to evaluate the limit of a function that gives a real number at that point? We aren't using the epsilon Delta definition, so there must be some other way. I guess I'm wondering how to evaluate a limit like this then.
– vevvvio
Nov 18 at 2:06
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
No. By "plugging in numbers," what you are assuming is continuity at the point you're plugging in. If you have already shown your function is continuous (or you are allowed to assume this), then you're fine, otherwise, you need to actually show that the limit equals the value you get by plugging in those numbers. The way one traditionally does this is by "epsilon-delta stuff."
answered Nov 18 at 0:14
Carl Schildkraut
10.6k11438
Right, but I had a quiz question that asked if a particular point was continuous, and this particular point gave a real number for its output in the function. I guess my struggle here is that to show a point is continuous, I must evaluate the limit of that point. How am I supposed to evaluate the limit of a function that gives a real number at that point? We aren't using the epsilon Delta definition, so there must be some other way. I guess I'm wondering how to evaluate a limit like this then.
– vevvvio
Nov 18 at 2:06
add a comment |
up vote
0
down vote
No. By "plugging in numbers," what you are assuming is continuity at the point you're plugging in. If you have already shown your function is continuous (or you are allowed to assume this), then you're fine, otherwise, you need to actually show that the limit equals the value you get by plugging in those numbers. The way one traditionally does this is by "epsilon-delta stuff."
answered Nov 18 at 0:14
Carl Schildkraut
10.6k11438
Right, but I had a quiz question that asked if a particular point was continuous, and this particular point gave a real number for its output in the function. I guess my struggle here is that to show a point is continuous, I must evaluate the limit of that point. How am I supposed to evaluate the limit of a function that gives a real number at that point? We aren't using the epsilon Delta definition, so there must be some other way. I guess I'm wondering how to evaluate a limit like this then.
– vevvvio
Nov 18 at 2:06
add a comment |
up vote
0
down vote
up vote
0
down vote
No. By "plugging in numbers," what you are assuming is continuity at the point you're plugging in. If you have already shown your function is continuous (or you are allowed to assume this), then you're fine, otherwise, you need to actually show that the limit equals the value you get by plugging in those numbers. The way one traditionally does this is by "epsilon-delta stuff."
answered Nov 18 at 0:14
Carl Schildkraut
10.6k11438
No. By "plugging in numbers," what you are assuming is continuity at the point you're plugging in. If you have already shown your function is continuous (or you are allowed to assume this), then you're fine, otherwise, you need to actually show that the limit equals the value you get by plugging in those numbers. The way one traditionally does this is by "epsilon-delta stuff."
answered Nov 18 at 0:14
Carl Schildkraut
10.6k11438
answered Nov 18 at 0:14
Carl Schildkraut
10.6k11438
answered Nov 18 at 0:14
Carl Schildkraut
10.6k11438
answered Nov 18 at 0:14
Carl Schildkraut
10.6k11438
10.6k11438
Right, but I had a quiz question that asked if a particular point was continuous, and this particular point gave a real number for its output in the function. I guess my struggle here is that to show a point is continuous, I must evaluate the limit of that point. How am I supposed to evaluate the limit of a function that gives a real number at that point? We aren't using the epsilon Delta definition, so there must be some other way. I guess I'm wondering how to evaluate a limit like this then.
– vevvvio
Nov 18 at 2:06
add a comment |
Right, but I had a quiz question that asked if a particular point was continuous, and this particular point gave a real number for its output in the function. I guess my struggle here is that to show a point is continuous, I must evaluate the limit of that point. How am I supposed to evaluate the limit of a function that gives a real number at that point? We aren't using the epsilon Delta definition, so there must be some other way. I guess I'm wondering how to evaluate a limit like this then.
– vevvvio
Nov 18 at 2:06
Right, but I had a quiz question that asked if a particular point was continuous, and this particular point gave a real number for its output in the function. I guess my struggle here is that to show a point is continuous, I must evaluate the limit of that point. How am I supposed to evaluate the limit of a function that gives a real number at that point? We aren't using the epsilon Delta definition, so there must be some other way. I guess I'm wondering how to evaluate a limit like this then.
– vevvvio
Nov 18 at 2:06
Right, but I had a quiz question that asked if a particular point was continuous, and this particular point gave a real number for its output in the function. I guess my struggle here is that to show a point is continuous, I must evaluate the limit of that point. How am I supposed to evaluate the limit of a function that gives a real number at that point? We aren't using the epsilon Delta definition, so there must be some other way. I guess I'm wondering how to evaluate a limit like this then.
– vevvvio
Nov 18 at 2:06
add a comment |
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