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Math25

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Mathematics
School
Cambridge University
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Homework
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This function is elementary, so it is continuous on its native domain. The only point which
is not in this domain is because we cannot divide by  It might be one more
point where division by zero occurs, i.e. if
  But this means

 which
is impossible.
The function has finite limits from the both sides of 
1) when


  and finally
2) when



 
and finally


We see that both limits exist and are finite but are different. This means that
is
discontinuous at and this type of discontinuity is called jump.
I think must be positive to use the sentence “-m<x<m”.
The function is elementary on each of three subintervals, so it is continuous on them. The
only points where a problem can arise are 
For the function to be continuous at limits from both sides must coincide, i.e.
 
This gives

Check what happens at  we need   


  which is equivalent to
  
  which is again true for 
The answer: 

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The vector equation
 
may be written as a system of three number equations,
each one for a corresponding coordinate:

 

 

From the third equation,
Substitute this to the second equation and obtain
 

Finally, substitute both
and
to the first equation and obtain


The answer: 


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This function is elementary, so it is continuous on its native domain. The only point which is not in this domain is 𝑥 = 0 because we cannot divide by 𝑥 = 0. It might be one more 1 1 point where division by zero occurs, i.e. if 𝑒 𝑥 − 1 = 0. But this means 𝑒 𝑥 = 1, 1 𝑥 = 0, which is impossible. The function has finite limits from the both sides of 𝑥 = 0: 1) when 𝑥 → 0+ , 1 → +∞, 𝑒 𝑥 → +∞, 𝑒 𝑥 − 1 → +∞ and finally 𝑓(𝑥) → 0+ ; 𝑥 1 1 2) when 𝑥 → 0− , 1 1 1 → −∞, 𝑒 𝑥 → 0+ , 𝑒 𝑥 − 1 → −1+ and finally 𝑓(𝑥) → −1− . 𝑥 We see that both limits exist and are finite but are different. This means that 𝑓(𝑥) is discontinuous at 𝑥 = 0 and this type of discontinuity is called jump. I think 𝑚 must be positive to use the sentence “-m ...
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Anonymous
Just what I was looking for! Super helpful.

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