How do you find the limit of # (x^3 - x) / (x -1)# as x approaches 1?

Answer 1
The limit can be evaluated by cancelling out #x - 1# as follows. If you notice that #x^3 - x# has common factors of #x#, factor out #x#.
#color(blue)(lim_(x->1) (x^3 - x)/(x - 1))#
#= lim_(x->1) (x(x^2 - 1))/(x - 1)#
Since #x^2 - 1# is a difference of two squares (#x^2 - a^2#, where #a# is a constant), you can factor this into #(x + 1)(x - 1)#.
#=> lim_(x->1) (x(x + 1)cancel((x - 1)))/cancel((x - 1))#
#= lim_(x->1) x(x + 1)#
Now you can just plug #x = 1# in.
#=> (1)(1 + 1) = color(blue)(2)#

And you can see from Wolfram Alpha that it is indeed correct.

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Answer 2

To find the limit of (x^3 - x) / (x - 1) as x approaches 1, we can simplify the expression by factoring the numerator. Factoring x^3 - x gives us x(x^2 - 1), which further simplifies to x(x - 1)(x + 1). Canceling out the common factor of (x - 1) in the numerator and denominator, we are left with x(x + 1). Now, substituting x = 1 into the simplified expression, we get 1(1 + 1) = 2. Therefore, the limit of (x^3 - x) / (x - 1) as x approaches 1 is equal to 2.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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