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How do you find the limit of #(x-3) /( x-4)# as x approaches #4^-#?

Answer 1

Ezra Adams

#lim_(x->4^-) (x-3)/(x-4) = -oo#

When #3 < x < 4# we have:

#0 < (x - 3) < 1#

#-1 < (x - 4) < 0#

So:

#(x-3)/(x-4) < 0#

since it is the quotient of a positive numerator and negative denominator.

When #x=4#, we have:

#(x-3)/(x-4) = 1/0#

Since this has a non-zero numerator and zero denominator, the function #(x-3)/(x-4)# has a vertical asymptote at #x=4# and we find:

#lim_(x->4^-) (x-3)/(x-4) = -oo#

graph{(x-3)/(x-4) [-7.88, 12.12, -4.16, 5.84]}

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Answer 2

Emilia Aguilar

To find the limit of (x-3)/(x-4) as x approaches 4^-, we substitute the value 4 into the expression. However, since the denominator becomes zero when x is 4, we need to approach the value from the left side.

By substituting x = 4 into the expression, we get (4-3)/(4-4) = 1/0, which is undefined. Therefore, the limit of (x-3)/(x-4) as x approaches 4^- does not exist.

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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.