Stuck on limits

Jacob

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Can someone please explain to me what’s happening to the numerator which I highlighted? It’s from my lecture but I don’t understand how it becomes sinx sinx sinh- cosx cosh sinx .
 

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Harry_the_cat

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I think it should be the other way around. That is, the negative of what you have.
 

Jacob

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I think it should be the other way around. That is, the negative of what you have.
^
Could you please elaborate more on what you mean by that? I know that the answer is -csc^2x but I want to know the working so it would be easier for me to master it but I'm stuck on the part that I highlighted. This is my first time dealing with cosecant, cotangent & secant
 
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Jacob

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I think it should be the other way around. That is, the negative of what you have.
In the lecture, my lecturer said that this 2 parts (which I marked x) will be canceled out. That makes sense but it didn’t when the next step he wrote the numerator as sinx sinx sinh - cosx cosh sinx.

Did I get it right?
 

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Subhotosh Khan

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Can someone please explain to me what’s happening to the numerator which I highlighted? It’s from my lecture but I don’t understand how it becomes sinx sinx sinh- cosx cosh sinx .
It does not! Do the algebra carefully!
 
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Subhotosh Khan

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In the lecture, my lecturer said that this 2 parts (which I marked x) will be canceled out. That makes sense but it didn’t when the next step he wrote the numerator as sinx sinx sinh - cosx cosh sinx.

Did I get it right?
No - I think it should be:

sin(x) * sin(x) * sin(h) - cos(x) * sin(h) * cos(x) ....,edited

Check it again......
 
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Jacob

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No - I think it should be:

sin(x) * sin(x) * sinh(x) - cos(x) * cosh(x) * cos(x)

Check it again......
This is exactly what my lecturer got. But I’m confused about how he got it. Why it becomes multiply? what happened to the negative sign in front of -sinx.

prob1.jpeg
 

Subhotosh Khan

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just writing the numerator of your attachment (Original)

sin(x) * [cos(x)*cos(h) - sin(x)*sin(h)] - cos(x) * [sin(x)*cos(h) + cos(x)*sin(h)] ..................................multiply and distribute

sin(x) * [cos(x)*cos(h)] - sin(x) * [sin(x)*sin(h)] - cos(x) * [sin(x)*cos(h)] - cos(x)*[cos(x)*sin(h)] .................. cancel appropriate terms

sin(x) * [cos(x)*cos(h)] - sin(x) * [sin(x)*sin(h)] - cos(x) * [sin(x)*cos(h)] - cos(x)*[cos(x)*sin(h)]

= - sin(x) * [sin(x)*sin(h)] - cos(x)*[cos(x)*sin(h)]

= -sin(h)

A little later I'll delete my previous responses.
 

Jacob

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just writing the numerator of your attachment (Original)

sin(x) * [cos(x)*cos(h) - sin(x)*sin(h)] - cos(x) * [sin(x)*cos(h) + cos(x)*sin(h)] ..................................multiply and distribute

sin(x) * [cos(x)*cos(h)] - sin(x) * [sin(x)*sin(h)] - cos(x) * [sin(x)*cos(h)] - cos(x)*[cos(x)*sin(h)] .................. cancel appropriate terms

sin(x) * [cos(x)*cos(h)] - sin(x) * [sin(x)*sin(h)] - cos(x) * [sin(x)*cos(h)] - cos(x)*[cos(x)*sin(h)]

= - sin(x) * [sin(x)*sin(h)] - cos(x)*[cos(x)*sin(h)]

= -sin(h)

A little later I'll delete my previous responses.
Thank you. It totally make sense right now
 

Subhotosh Khan

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Can someone please explain to me what’s happening to the numerator which I highlighted? It’s from my lecture but I don’t understand how it becomes sinx sinx sinh- cosx cosh sinx .
Now I hope you see that it is NOT sinx sinx sinh- cosx cosh sinx .........
 

Jacob

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Now I hope you see that it is NOT sinx sinx sinh- cosx cosh sinx .........
Yes finally, thanks to you. May I know how you get

= - sin(x) * [sin(x)*sin(h)] - cos(x)*[cos(x)*sin(h)] (This part right here become -sin(h)?

= -sin(h)
 

Subhotosh Khan

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Yes finally, thanks to you. May I know how you get

= - sin(x) * [sin(x)*sin(h)] - cos(x)*[cos(x)*sin(h)] (This part right here become -sin(h)?

= -sin(h)
- sin(x) * [sin(x)*sin(h)] - cos(x)*[cos(x)*sin(h)]

= -sin(x)*sin(x)*sin(h) - cos(x) *cos(x)*sin(h)

= -sin2(x) * sin(h) - cos2(x) * sin(h)

= - [sin2(x) + cos2(x)] * sin(h)

= - [1] * sin(h)

= - sin(h)

Please work with pencil and paper instead of staring at the screen!!
 

Jacob

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- sin(x) * [sin(x)*sin(h)] - cos(x)*[cos(x)*sin(h)]

= -sin(x)*sin(x)*sin(h) - cos(x) *cos(x)*sin(h)

= -sin2(x) * sin(h) - cos2(x) * sin(h)

= - [sin2(x) + cos2(x)] * sin(h)

= - [1] * sin(h)

= - sin(h)

Please work with pencil and paper instead of staring at the screen!!
Sorry, I did work through the problem. But this part = - [sin2(x) + cos2(x)] * sin(h) I got -ve instead +ve. May i know how it is +ve

prob.jpeg
 
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