Science:Math Exam Resources/Courses/MATH110/December 2012/Question 02 (b)

MATH110 December 2012

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Question 02 (b)
Determine whether the following statement is true or false. If it is true, provide justification. If it is false, provide a counterexample. $\displaystyle \lim _{x\rightarrow 2}{\dfrac {2-x}{4-x^{2}}}={\dfrac {1}{4}}$.

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Try simplifying the limit on the left hand side of the equal sign. How can we eliminate the issue with the denominator tending to 0?

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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. As suggested in the hint, we will simplify the limit on the left hand side of the equal sign. If we just plug in 2 we get 0/0, so we will have to do some simplifying. Because the numerator and denominator are polynomials, factoring is a good first step. The denominator is a difference of squares, so we can write: $\lim _{x\to 2}{\frac {x-2}{4-x^{2}}}=\lim _{x\to 2}{\frac {2-x}{(2-x)(2+x)}}$ Canceling the common terms, and then plugging in the 2 gives: $\lim _{x\to 2}{\frac {1}{2+x}}={\frac {1}{2+2}}={\frac {1}{4}}$ Which matches the right hand side of the original equation. Thus the statement is true.

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