Evaluating $lim_{hto 0}frac{1}{h}(,(x+h)sec(x+h) - x sec(x),)$ without L'Hôpital's rule [closed]
$begingroup$
I got this problem as a high school problem. I want to find
$$lim_{hto 0}frac{(x+h)sec(x+h) - xsec(x)}{h}$$
I know that $$lim_{xto 0}frac{sin x}{x}=1qquadtext{and}qquadlim_{xto 0}frac{1 - cos x}{x}=0qquadtext{and}qquadlim_{xto 0}frac{tan x}{x}=1$$
I need to solve it using mainly these three standard forms.
limits trigonometry limits-without-lhopital
edited Jan 8 at 19:41
Blue
48k870153
asked Jan 8 at 16:15
user630305user630305
63
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closed as off-topic by Saad, MisterRiemann, TravisJ, Siong Thye Goh, Alexander Gruber♦ Jan 8 at 23:13
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, MisterRiemann, TravisJ, Siong Thye Goh, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I got this problem as a high school problem. I want to find
$$lim_{hto 0}frac{(x+h)sec(x+h) - xsec(x)}{h}$$
I know that $$lim_{xto 0}frac{sin x}{x}=1qquadtext{and}qquadlim_{xto 0}frac{1 - cos x}{x}=0qquadtext{and}qquadlim_{xto 0}frac{tan x}{x}=1$$
I need to solve it using mainly these three standard forms.
limits trigonometry limits-without-lhopital
edited Jan 8 at 19:41
Blue
48k870153
asked Jan 8 at 16:15
user630305user630305
63
$endgroup$
closed as off-topic by Saad, MisterRiemann, TravisJ, Siong Thye Goh, Alexander Gruber♦ Jan 8 at 23:13
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, MisterRiemann, TravisJ, Siong Thye Goh, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Note that this is, by definition, $left(x sec xright)'$.
$endgroup$
– StackTD
Jan 8 at 16:29
$begingroup$
I want to find what work you did. Do you know the derivatives of sine and cosine? The $(sin h)/h$ limit as $h$ goes to zero?
$endgroup$
– Oscar Lanzi
Jan 8 at 16:30
$begingroup$
@OscarLanzi Yes I know
$endgroup$
– user630305
Jan 8 at 16:43
add a comment |
$begingroup$
I got this problem as a high school problem. I want to find
$$lim_{hto 0}frac{(x+h)sec(x+h) - xsec(x)}{h}$$
I know that $$lim_{xto 0}frac{sin x}{x}=1qquadtext{and}qquadlim_{xto 0}frac{1 - cos x}{x}=0qquadtext{and}qquadlim_{xto 0}frac{tan x}{x}=1$$
I need to solve it using mainly these three standard forms.
limits trigonometry limits-without-lhopital
edited Jan 8 at 19:41
Blue
48k870153
asked Jan 8 at 16:15
user630305user630305
63
$endgroup$
I got this problem as a high school problem. I want to find
$$lim_{hto 0}frac{(x+h)sec(x+h) - xsec(x)}{h}$$
I know that $$lim_{xto 0}frac{sin x}{x}=1qquadtext{and}qquadlim_{xto 0}frac{1 - cos x}{x}=0qquadtext{and}qquadlim_{xto 0}frac{tan x}{x}=1$$
I need to solve it using mainly these three standard forms.
limits trigonometry limits-without-lhopital
limits trigonometry limits-without-lhopital
edited Jan 8 at 19:41
Blue
48k870153
asked Jan 8 at 16:15
user630305user630305
63
edited Jan 8 at 19:41
Blue
48k870153
asked Jan 8 at 16:15
user630305user630305
63
edited Jan 8 at 19:41
Blue
48k870153
edited Jan 8 at 19:41
Blue
48k870153
edited Jan 8 at 19:41
Blue
48k870153
48k870153
asked Jan 8 at 16:15
user630305user630305
63
asked Jan 8 at 16:15
user630305user630305
63
asked Jan 8 at 16:15
user630305user630305
63
63
closed as off-topic by Saad, MisterRiemann, TravisJ, Siong Thye Goh, Alexander Gruber♦ Jan 8 at 23:13
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, MisterRiemann, TravisJ, Siong Thye Goh, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Saad, MisterRiemann, TravisJ, Siong Thye Goh, Alexander Gruber♦ Jan 8 at 23:13
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, MisterRiemann, TravisJ, Siong Thye Goh, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Note that this is, by definition, $left(x sec xright)'$.
$endgroup$
– StackTD
Jan 8 at 16:29
$begingroup$
I want to find what work you did. Do you know the derivatives of sine and cosine? The $(sin h)/h$ limit as $h$ goes to zero?
$endgroup$
– Oscar Lanzi
Jan 8 at 16:30
$begingroup$
@OscarLanzi Yes I know
$endgroup$
– user630305
Jan 8 at 16:43
add a comment |
$begingroup$
Note that this is, by definition, $left(x sec xright)'$.
$endgroup$
– StackTD
Jan 8 at 16:29
$begingroup$
I want to find what work you did. Do you know the derivatives of sine and cosine? The $(sin h)/h$ limit as $h$ goes to zero?
$endgroup$
– Oscar Lanzi
Jan 8 at 16:30
$begingroup$
@OscarLanzi Yes I know
$endgroup$
– user630305
Jan 8 at 16:43
$begingroup$
Note that this is, by definition, $left(x sec xright)'$.
$endgroup$
– StackTD
Jan 8 at 16:29
$begingroup$
Note that this is, by definition, $left(x sec xright)'$.
$endgroup$
– StackTD
Jan 8 at 16:29
$begingroup$
I want to find what work you did. Do you know the derivatives of sine and cosine? The $(sin h)/h$ limit as $h$ goes to zero?
$endgroup$
– Oscar Lanzi
Jan 8 at 16:30
$begingroup$
I want to find what work you did. Do you know the derivatives of sine and cosine? The $(sin h)/h$ limit as $h$ goes to zero?
$endgroup$
– Oscar Lanzi
Jan 8 at 16:30
$begingroup$
@OscarLanzi Yes I know
$endgroup$
– user630305
Jan 8 at 16:43
$begingroup$
@OscarLanzi Yes I know
$endgroup$
– user630305
Jan 8 at 16:43
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
$$dfrac{(x+h)cos x-xcos(x+h)}{hcos xcos(x+h)}$$
$$=dfrac1{cos xcos(x+h)}left(xcdotdfrac{cos x-cos(x+h)}{h}+cos xright)$$
Now use http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html
answered Jan 8 at 17:31
lab bhattacharjeelab bhattacharjee
225k15156274
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add a comment |
$begingroup$
Maybe try something like this. :-)
$$lim_{hto0}frac{(x+h)sec(x+h)-xsec x}{h}$$
$$= lim_{hto0}frac{(x+h)sec(x+h)-xsec(x+h) +xsec(x+h)-xsec x}{h}$$
answered Jan 8 at 16:30
John JoyJohn Joy
6,20611526
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add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$$dfrac{(x+h)cos x-xcos(x+h)}{hcos xcos(x+h)}$$
$$=dfrac1{cos xcos(x+h)}left(xcdotdfrac{cos x-cos(x+h)}{h}+cos xright)$$
Now use http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html
answered Jan 8 at 17:31
lab bhattacharjeelab bhattacharjee
225k15156274
$endgroup$
add a comment |
$begingroup$
$$dfrac{(x+h)cos x-xcos(x+h)}{hcos xcos(x+h)}$$
$$=dfrac1{cos xcos(x+h)}left(xcdotdfrac{cos x-cos(x+h)}{h}+cos xright)$$
Now use http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html
answered Jan 8 at 17:31
lab bhattacharjeelab bhattacharjee
225k15156274
$endgroup$
add a comment |
$begingroup$
$$dfrac{(x+h)cos x-xcos(x+h)}{hcos xcos(x+h)}$$
$$=dfrac1{cos xcos(x+h)}left(xcdotdfrac{cos x-cos(x+h)}{h}+cos xright)$$
Now use http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html
answered Jan 8 at 17:31
lab bhattacharjeelab bhattacharjee
225k15156274
$endgroup$
$$dfrac{(x+h)cos x-xcos(x+h)}{hcos xcos(x+h)}$$
$$=dfrac1{cos xcos(x+h)}left(xcdotdfrac{cos x-cos(x+h)}{h}+cos xright)$$
Now use http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html
answered Jan 8 at 17:31
lab bhattacharjeelab bhattacharjee
225k15156274
answered Jan 8 at 17:31
lab bhattacharjeelab bhattacharjee
225k15156274
answered Jan 8 at 17:31
lab bhattacharjeelab bhattacharjee
225k15156274
answered Jan 8 at 17:31
lab bhattacharjeelab bhattacharjee
225k15156274
225k15156274
add a comment |
add a comment |
$begingroup$
Maybe try something like this. :-)
$$lim_{hto0}frac{(x+h)sec(x+h)-xsec x}{h}$$
$$= lim_{hto0}frac{(x+h)sec(x+h)-xsec(x+h) +xsec(x+h)-xsec x}{h}$$
answered Jan 8 at 16:30
John JoyJohn Joy
6,20611526
$endgroup$
add a comment |
$begingroup$
Maybe try something like this. :-)
$$lim_{hto0}frac{(x+h)sec(x+h)-xsec x}{h}$$
$$= lim_{hto0}frac{(x+h)sec(x+h)-xsec(x+h) +xsec(x+h)-xsec x}{h}$$
answered Jan 8 at 16:30
John JoyJohn Joy
6,20611526
$endgroup$
add a comment |
$begingroup$
Maybe try something like this. :-)
$$lim_{hto0}frac{(x+h)sec(x+h)-xsec x}{h}$$
$$= lim_{hto0}frac{(x+h)sec(x+h)-xsec(x+h) +xsec(x+h)-xsec x}{h}$$
answered Jan 8 at 16:30
John JoyJohn Joy
6,20611526
$endgroup$
Maybe try something like this. :-)
$$lim_{hto0}frac{(x+h)sec(x+h)-xsec x}{h}$$
$$= lim_{hto0}frac{(x+h)sec(x+h)-xsec(x+h) +xsec(x+h)-xsec x}{h}$$
answered Jan 8 at 16:30
John JoyJohn Joy
6,20611526
edited Jan 8 at 18:26
answered Jan 8 at 16:30
John JoyJohn Joy
6,20611526
answered Jan 8 at 16:30
John JoyJohn Joy
6,20611526
answered Jan 8 at 16:30
John JoyJohn Joy
6,20611526
6,20611526
add a comment |
add a comment |
$begingroup$
Note that this is, by definition, $left(x sec xright)'$.
$endgroup$
– StackTD
Jan 8 at 16:29
$begingroup$
I want to find what work you did. Do you know the derivatives of sine and cosine? The $(sin h)/h$ limit as $h$ goes to zero?
$endgroup$
– Oscar Lanzi
Jan 8 at 16:30
$begingroup$
@OscarLanzi Yes I know
$endgroup$
– user630305
Jan 8 at 16:43