How do you determine the limit of #(2)(/(x-5)^(3))# as x approaches #5^-#??

Answer 1

Charles Arocha

#= -oo#

#lim_(x to 5^-) 2/(x-5)^3#

let #x = 5 - h, 0< h " << " 1#

#lim_(x to 5^-) 2/(x-5)^3 implies lim_(h to 0) 2/(-h)^3#

#implies lim_(h to 0) - 2/h^3 = -oo# as #h > 0#

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Answer 2

Isaiah Allen

To determine the limit of (2)/(x-5)^3 as x approaches 5^-, we can substitute the value of x into the expression and simplify.

Substituting x = 5^- into the expression, we get:

(2)/(5^- - 5)^3

Simplifying further:

(2)/((-5)^-3)

Using the rule that a negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent:

(2)/((-1/5)^3)

Cubing the reciprocal of -1/5:

(2)/(-1/125)

Multiplying by the reciprocal of -1/125 is the same as multiplying by -125/1:

(2) * (-125/1)

Simplifying the multiplication:

-250

Therefore, the limit of (2)/(x-5)^3 as x approaches 5^- is -250.

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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.