How do you find the limit of #(x^2-3x+2)/(x^3-4x)# as x approaches 2 from the right, as x approaches -2 from the right, as x approaches 0 from the left, and as x approaches 1 from the right?

Question

Cameron Alexander · Accepted Answer

To begin, factor the expression. #{x^2-3x+2}/{x^3-4x}={(x-2)(x-1)}/{x(x+2)(x-2)}={(x-1)}/{x(x+2)}# Start taking the necessary limits now. #lim_{x \rarr0^-}[{(x-1)}/{x(x+2)}]# #=lim_{x \rarr0^-}[{(x-1)}/{(x+2)}]*lim_{x \rarr0^-}[1/x]# #={(0-1)}/{(0+2)}*(-\infty)=-infty# #lim_{x \rarr0^-}[1/x]=-infty# because we are approaching zero from the negative side of the number line.  Regarding the following boundary, #lim_{x \rarr-2^+}[{(x-1)}/{x(x+2)}]# #=lim_{x \rarr-2^+}[{(x-1)}/{x}]*lim_{x \rarr-2^+}[1/{(x+2)}]# #={-3}/{-2}*(+infty)=+infty# #lim_{x \rarr-2^+}[1/{(x+2)}]=+infty# because as #(x\rarr-2^+)#, #(x+2)# gets very small, but stays positive. Regarding the following boundary, #lim_{x \rarr 1^+}[{(x-1)}/{x(x+2)}]=lim_{x \rarr1^+}[{1}/{x(x+2)}]lim_{x \rarr1^+}[(x-1)]#
#=1/{1*3}*0=0# Regarding the final limit, #lim_{x \rarr2^+}[ {(x-1)}/{x(x+2)}]=3/{2*4}=3/8# Even though the denominator of the original expression went to zero at #x=2#, the limit is still finite because the numerator went to zero just as quickly.  Recall, at the start we were able to cross out the #(x-2)# in the denominator (which was causing the singularity) with another #(x-2)# factor in the numerator.

Axel Alvarez · Answer

undefined To find the limit of (x^2-3x+2)/(x^3-4x) as x approaches 2 from the right, we substitute 2 into the expression and simplify. The limit is 1/4.

To find the limit as x approaches -2 from the right, we substitute -2 into the expression and simplify. The limit is -1/6.

To find the limit as x approaches 0 from the left, we substitute 0 into the expression and simplify. The limit is -1/0, which is undefined.

To find the limit as x approaches 1 from the right, we substitute 1 into the expression and simplify. The limit is 0.

Luke Alvarez · Answer

undefined As x approaches 2 from the right, the limit of the function is 2/4 = 1/2.

As x approaches -2 from the right, the limit of the function is 6/0, which is undefined.

As x approaches 0 from the left, the limit of the function is 2/0, which is undefined.

As x approaches 1 from the right, the limit of the function is 0/3 = 0.