Science:Math Exam Resources/Courses/MATH110/December 2017/Question 03 (a)

MATH110 December 2017

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Question 03 (a)
Evaluate the following limits or determine they are either infinite or do not exist. (a) ${\textstyle \displaystyle \lim _{x\to 1}x{\sqrt {x}}+1}$

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
One can plug in values to a limit provided the function is continuous.

Hint 2
A power function is continuous, whenever it is defined.

Hint 3

Strictly speaking, the limit means $$\left(\lim _{x\to 1}x{\sqrt {x}}\right)+1,$$ as opposed to $$\lim _{x\to 1}\left(x{\sqrt {x}}+1\right).$$ However, these two are equivalent since the constant function ${\textstyle 1}$ is continuous and has limit ${\textstyle 1}$ (trivially!) when ${\textstyle x=1}$. So one of these limits exist if and only if the other is, and are equal provided either exists. In other words, there is no ambiguity in these two possible interpretations.

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. By the rule of exponents, one can rewrite $$\lim _{x\to 1}x{\sqrt {x}}+1=\lim _{x\to 1}x\cdot {x}^{\frac {1}{2}}+1=\lim _{x\to 1}x^{\frac {3}{2}}+1.$$ Since the power function ${\textstyle x^{\frac {3}{2}}}$ is continuous at the point ${\textstyle x=1}$, we have $$\lim _{x\to 1}x^{\frac {3}{2}}=1^{\frac {3}{2}}=1.$$ Therefore, $$\lim _{x\to 1}x{\sqrt {x}}+1=\lim _{x\to 1}x^{\frac {3}{2}}+1=1+1=2.$$ Answer: ${\textstyle \lim \limits _{x\to 1}x{\sqrt {x}}+1={\color {blue}2}}$.