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Question 10 

Let be a real number, and let
For what values is the function continuous at ? Justify your answer. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Split into different cases for . 
Hint 2 

Split into different cases for , and . 
Hint 3 

Use the Squeeze Theorem to check for . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. In order to be continuous at we need So it remains to determine which gives
So by the Squeeze Theorem, as . Thus is continuous at when . 