The problem and my work towards solving it are attached. But not very sure about the answer. Can anybody please help.
For interested readers, the text in the images appears to be as follows:
4. For what value(s) of a is the following limit positive?
. . . . .\(\displaystyle \displaystyle \lim_{x \rightarrow 1}\, \dfrac{x^2\, -\, 2x\, +\, a}{2x^2\, -\, x\, -\, 1}\)
\(\displaystyle \displaystyle \lim_{x \rightarrow 1}\, \dfrac{x^2\, -\, 2x\, +\, a}{2x^2\, -\, x\, -\, 1}\, >\, 0\)
\(\displaystyle \mbox{Factorize den }\, \longrightarrow\, (2x\, +\, 1)(x\, -\, 1)\)
\(\displaystyle (x^2\, -\, 2x\, +\, a)\, (2x\, +\, 1)\, (x\, -\, 1)\, >\, 0\)
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. . . . . . .\(\displaystyle [(4x^2\, -\, 1)\, (x^2\, -\, 1)]\, \nearrow\)
\(\displaystyle (x^2\, -\, 2x\, +\, a)\, (2x\, +\, 1)\, (x\, -\, 1)\, >\, 0\)
. . . . . . . . . . . . . . .\(\displaystyle \searrow\, \swarrow\)
. . . . . . . . . . . . .\(\displaystyle \mbox{both}\, >\, 0\)
\(\displaystyle \therefore (x^2\, -\, 2x\, +\, a)\, >\, 0\)
\(\displaystyle \mbox{sub lim }\, x\, \rightarrow\, 1\)
\(\displaystyle 1\, -\, 2\, +\, a\, >\, 0\)
\(\displaystyle -1\, +\, a\, >\, 0\)
\(\displaystyle \underline{ a\, >\, 1 }\)
Original poster: Please reply with corrections or confirmation. Thank you!