One-sided limits for piecewise fcns: F(x)=(x^3+x^2+2x+1)/(x+1) for x != 1, -4 for x =

[Markx14]

Find the limit of F(x) as x approaches 1+ and 1- for the following functions:

. . .\(\displaystyle \mbox{a. }\, F(x)\, =\, \begin{cases} \dfrac{x^3\, +\, x^2\, +\, 2x\, +\, 1}{x\, +\, 1} & \mbox{for }\, x\, \neq\, 1 \\ \, & \, \\ -4 & \mbox{for }\, x\, =\, 1 \end{cases}\)

. . .\(\displaystyle \mbox{b. }\, F(x)\, =\, \begin{cases} \dfrac{\sqrt{\strut x\,}\, -\, 1}{x\, -\, \sqrt{\strut 2\, -\, x\,}} & \mbox{for }\, x\, \neq\, 1 \\ \, & \, \\ 2 & \mbox{for }\, x\, =\, 1 \end{cases}\)

 

[Deleted member 4993]

Find the limit of F(x) as x approaches 1+ and 1- for the following functions:

. . .\(\displaystyle \mbox{a. }\, F(x)\, =\, \begin{cases} \dfrac{x^3\, +\, x^2\, +\, 2x\, +\, 1}{x\, +\, 1} & \mbox{for }\, x\, \neq\, 1 \\ \, & \, \\ -4 & \mbox{for }\, x\, =\, 1 \end{cases}\)

. . .\(\displaystyle \mbox{b. }\, F(x)\, =\, \begin{cases} \dfrac{\sqrt{\strut x\,}\, -\, 1}{x\, -\, \sqrt{\strut 2\, -\, x\,}} & \mbox{for }\, x\, \neq\, 1 \\ \, & \, \\ 2 & \mbox{for }\, x\, =\, 1 \end{cases}\)

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[HallsofIvy]

First, of course, the limit "as x goes to 1" does not depend upon the value at x= 1 so, for the first, you are only concerned with \(\displaystyle \frac{x^3+ x^2+ 2x+ 1}{x+ 1}\). That should be trivial. (It is important to notice that 1+1= 2 not 0!)

Likewise, for the second, you really only need to look at \(\displaystyle \frac{\sqrt{x}- 1}{x- \sqrt{2- x}}\). I suggest "rationalizing the denominator" by multiplying both numerator and denominator by \(\displaystyle x+ \sqrt{2- x}\).