MATH103 April 2015
• Q1 (a) (i) • Q1 (a) (ii) • Q1 (a) (iii) • Q1 (b) (i) • Q1 (b) (ii) • Q1 (b) (iii) • Q1 (c) (i) • Q1 (c) (ii) • Q1 (c) (iii) • Q1 (d) (i) • Q1 (d) (ii) • Q1 (e) (i) • Q1 (e) (ii) • Q1 (e) (iii) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q2 (e) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 (a) • Q7 (b) • Q7 (c) • Q7 (d) • Q8 • Q9 • Q10 (a) • Q10 (b) •
Question 02 (c)
Evaluate the following limit: .
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.
Use the L'hospital's rule.
To get the derivative of the numerator, use fundamental theorem of calculus.
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.
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Let and .
At , we can easily observe that . Therefore, we use L'hospital's rule.
For this purpose, we need to calculate the derivatives of and at 0. By fundamental theorem of calculus, we have
and hence .
On the other hand, we have , so .
Therefore, we apply L'Hospital's rule again, so that we consider their second derivatives:
Therefore, collecting these, we find the limit as