Question

Does anyone know why the function f(x)= 1/(x-1)^4 the limit as it approaches to 1 is infinite. I thought that in this case we should use the one-side limits because when x=1 the denominator will be 0.

Answer 1

if you get the same value for both the left and right-sided limit, then that value is the limit.
In this case
approaching from the left side of 1 (below 1)
\[\lim_{x \rightarrow 1^-} \frac{ 1 }{ (x-1)^4 }=\frac{ 1 }{ 0 }= \infty\]
notice that though 0.999-1.000 = -0.001 is negative, after raising to an even power (4 in this case) we get a positive number that approaches 0
similarly, the other side also gives
\[\lim_{x \rightarrow 1^+} \frac{ 1 }{ (x-1)^4 }=\frac{ 1 }{ 0 }= \infty\]
example: 1.001 -1.000= 0.001, and raised to the 4th power, is a small positive number that approaches 0, and consequently the fraction approaches infinity.