Limits of combined functions Problem (Khan Academy)
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PROBLEM: Limits do not exist for both of the functions as $x to -2$ but they say both functions still have fixed limit value equal to $5$. Is it one of the Limits Rules or more of common-sense approach ?
As you can see, for both $f(x), g(x)$ have different values for limits when approached from left and right sides, which mean they don't have a limit by definition. But if we add them together and take their limit it is $5$ for both left and right side. Hence, Khan Academy concludes this composite function does have a limit.
Can I assume that this is what it means:
Limits Rule for composite functions: if limits do not exist for 2 or more individual functions but added together they reach same value for both left-side and right-side limits, then limit exists.
limits
asked Aug 13 at 9:07
Arnuld
5011
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up vote
0
down vote
favorite
PROBLEM: Limits do not exist for both of the functions as $x to -2$ but they say both functions still have fixed limit value equal to $5$. Is it one of the Limits Rules or more of common-sense approach ?
As you can see, for both $f(x), g(x)$ have different values for limits when approached from left and right sides, which mean they don't have a limit by definition. But if we add them together and take their limit it is $5$ for both left and right side. Hence, Khan Academy concludes this composite function does have a limit.
Can I assume that this is what it means:
Limits Rule for composite functions: if limits do not exist for 2 or more individual functions but added together they reach same value for both left-side and right-side limits, then limit exists.
limits
asked Aug 13 at 9:07
Arnuld
5011
Try drawing a graph of $g(x)+h(x)$ to see how it behaves. Recall that the limit of a function does not depend on how you describe it, but only on what the function value is at each point.
– Henning Makholm
Aug 13 at 9:17
Thanks. Good idea to graph it, did not occur to me :)
– Arnuld
Aug 13 at 10:19
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
PROBLEM: Limits do not exist for both of the functions as $x to -2$ but they say both functions still have fixed limit value equal to $5$. Is it one of the Limits Rules or more of common-sense approach ?
As you can see, for both $f(x), g(x)$ have different values for limits when approached from left and right sides, which mean they don't have a limit by definition. But if we add them together and take their limit it is $5$ for both left and right side. Hence, Khan Academy concludes this composite function does have a limit.
Can I assume that this is what it means:
Limits Rule for composite functions: if limits do not exist for 2 or more individual functions but added together they reach same value for both left-side and right-side limits, then limit exists.
limits
asked Aug 13 at 9:07
Arnuld
5011
PROBLEM: Limits do not exist for both of the functions as $x to -2$ but they say both functions still have fixed limit value equal to $5$. Is it one of the Limits Rules or more of common-sense approach ?
As you can see, for both $f(x), g(x)$ have different values for limits when approached from left and right sides, which mean they don't have a limit by definition. But if we add them together and take their limit it is $5$ for both left and right side. Hence, Khan Academy concludes this composite function does have a limit.
Can I assume that this is what it means:
Limits Rule for composite functions: if limits do not exist for 2 or more individual functions but added together they reach same value for both left-side and right-side limits, then limit exists.
limits
asked Aug 13 at 9:07
Arnuld
5011
asked Aug 13 at 9:07
Arnuld
5011
asked Aug 13 at 9:07
Arnuld
5011
asked Aug 13 at 9:07
Arnuld
5011
5011
Try drawing a graph of $g(x)+h(x)$ to see how it behaves. Recall that the limit of a function does not depend on how you describe it, but only on what the function value is at each point.
– Henning Makholm
Aug 13 at 9:17
Thanks. Good idea to graph it, did not occur to me :)
– Arnuld
Aug 13 at 10:19
add a comment |Â
Try drawing a graph of $g(x)+h(x)$ to see how it behaves. Recall that the limit of a function does not depend on how you describe it, but only on what the function value is at each point.
– Henning Makholm
Aug 13 at 9:17
Thanks. Good idea to graph it, did not occur to me :)
– Arnuld
Aug 13 at 10:19
Try drawing a graph of $g(x)+h(x)$ to see how it behaves. Recall that the limit of a function does not depend on how you describe it, but only on what the function value is at each point.
– Henning Makholm
Aug 13 at 9:17
Try drawing a graph of $g(x)+h(x)$ to see how it behaves. Recall that the limit of a function does not depend on how you describe it, but only on what the function value is at each point.
– Henning Makholm
Aug 13 at 9:17
Thanks. Good idea to graph it, did not occur to me :)
– Arnuld
Aug 13 at 10:19
Thanks. Good idea to graph it, did not occur to me :)
– Arnuld
Aug 13 at 10:19
add a comment |Â
1 Answer
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Let $f(x)=g(x)+h(x)$, what we are interested would be the function $f$, rather than function $g$ or $h$.
We can then study $lim_x to c^-f(x)$ and $lim_x to c^+f(x)$ to decide if the limit exists at $c$. Limit need not exists at point $c$ for $g$ and $h$.
Remark:
The word composite usually refers to $fcirc g$, then that $fcirc g(x)=f(g(x))$.
answered Aug 13 at 9:13
Siong Thye Goh
78.8k134997
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Let $f(x)=g(x)+h(x)$, what we are interested would be the function $f$, rather than function $g$ or $h$.
We can then study $lim_x to c^-f(x)$ and $lim_x to c^+f(x)$ to decide if the limit exists at $c$. Limit need not exists at point $c$ for $g$ and $h$.
Remark:
The word composite usually refers to $fcirc g$, then that $fcirc g(x)=f(g(x))$.
answered Aug 13 at 9:13
Siong Thye Goh
78.8k134997
add a comment |Â
up vote
1
down vote
accepted
Let $f(x)=g(x)+h(x)$, what we are interested would be the function $f$, rather than function $g$ or $h$.
We can then study $lim_x to c^-f(x)$ and $lim_x to c^+f(x)$ to decide if the limit exists at $c$. Limit need not exists at point $c$ for $g$ and $h$.
Remark:
The word composite usually refers to $fcirc g$, then that $fcirc g(x)=f(g(x))$.
answered Aug 13 at 9:13
Siong Thye Goh
78.8k134997
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Let $f(x)=g(x)+h(x)$, what we are interested would be the function $f$, rather than function $g$ or $h$.
We can then study $lim_x to c^-f(x)$ and $lim_x to c^+f(x)$ to decide if the limit exists at $c$. Limit need not exists at point $c$ for $g$ and $h$.
Remark:
The word composite usually refers to $fcirc g$, then that $fcirc g(x)=f(g(x))$.
answered Aug 13 at 9:13
Siong Thye Goh
78.8k134997
Let $f(x)=g(x)+h(x)$, what we are interested would be the function $f$, rather than function $g$ or $h$.
We can then study $lim_x to c^-f(x)$ and $lim_x to c^+f(x)$ to decide if the limit exists at $c$. Limit need not exists at point $c$ for $g$ and $h$.
Remark:
The word composite usually refers to $fcirc g$, then that $fcirc g(x)=f(g(x))$.
answered Aug 13 at 9:13
Siong Thye Goh
78.8k134997
edited Aug 13 at 9:21
answered Aug 13 at 9:13
Siong Thye Goh
78.8k134997
answered Aug 13 at 9:13
Siong Thye Goh
78.8k134997
answered Aug 13 at 9:13
Siong Thye Goh
78.8k134997
78.8k134997
add a comment |Â
add a comment |Â
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Try drawing a graph of $g(x)+h(x)$ to see how it behaves. Recall that the limit of a function does not depend on how you describe it, but only on what the function value is at each point.
– Henning Makholm
Aug 13 at 9:17
Thanks. Good idea to graph it, did not occur to me :)
– Arnuld
Aug 13 at 10:19