How do you evaluate # (x^2-16) /( x^2+3x-4)# as x approaches 1+?

Answer 1

Please see the explanation section, below.

First note that #lim_(xrarr1^+) (x^2-16) = -15#. (The limit of the numerator is not #0#.)
Furthermore #lim_(xrarr1^+) (x^2+3x-4) = 0# (The limit of the denominator is #0#.)
The form of this limit is #("non-"0)/0#, there is no limit (the limit does not exist).
But we can say more about why the limit does not exist. The one-sided limit does noty exist because the function is either increasing without bound or it is decreasing without bound as #xrarr1^+#. Let's figure out which.
The numerator is approaching a negative number while the denominator goes to #0# at #1#. This tells us that #x-1# is a faxctor of the deniminator.
#x^2+3x-4 = (x-1)(x+4)#
There is one factor approaching #0# and another approaching #5#. Since #xrarr1^+#, we have #x > 1#, so #x-1 > 0#. Therefore the denominator is approaching #0#, and is positive.

A negative numerator and a positive denominator give us negative values.

The quotient does not approach a limit because, as #x# approaches #1# from the right, the function decreases without bound.

We abbreviate the previous sentence by writing

#lim_(xrarr1^+) (x^2-16)/(x^2+3x-4) = -oo#.
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Answer 2

To evaluate the expression (x^2-16) / (x^2+3x-4) as x approaches 1+, we substitute the value of x into the expression.

(x^2-16) / (x^2+3x-4) = ((1+)^2-16) / ((1+)^2+3(1+)-4)

Simplifying further:

((1+)^2-16) / ((1+)^2+3(1+)-4) = (1-16) / (1+3+3-4)

Calculating:

(-15) / (3) = -5

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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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