How do you find the limit of #(x^3 *abs(2x-6))/ (x-3)# as x approaches 3?

Answer 1

The one sided limits disagree, so there is no two sided limit.

Note that if #t >= 0# then #abs(t) = t# and if #t <= 0# then #abs(t) = -t#.
When #x > 3# then #x - 3 > 0# and we have:
#(x^3*abs(2x-6))/(x-3) = x^3*abs(2(x-3))/(x-3) = (x^3*2(color(red)(cancel(color(black)(x-3)))))/(color(red)(cancel(color(black)(x-3)))) = 2x^3#
Hence #lim_(x->3^+) (x^3*abs(2x-6))/(x-3) = lim_(x->3^+) 2x^3 = 54#
When #x < 3# then #x - 3 < 0# and we have:
#(x^3*abs(2x-6))/(x-3) = x^3*abs(2(x-3))/(x-3) = (x^3*(-2(color(red)(cancel(color(black)(x-3))))))/(color(red)(cancel(color(black)(x-3)))) = -2x^3#
Hence #lim_(x->3^-) (x^3*abs(2x-6))/(x-3) = lim_(x->3^-) -2x^3 = -54#
Since the one sided limits disagree there is no two sided limit as #x->3#.
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Answer 2

To find the limit of the given expression as x approaches 3, we can use algebraic manipulation and the properties of limits. First, we simplify the expression by factoring out common terms:

(x^3 * abs(2x-6))/ (x-3) = (x^3 * abs(2(x-3)))/ (x-3) = x^3 * abs(2)

Next, we can evaluate the limit separately for each term. As x approaches 3, the absolute value term remains constant at 2. The limit of x^3 as x approaches 3 is 3^3 = 27.

Therefore, the limit of the given expression as x approaches 3 is 27 * 2 = 54.

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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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