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How do you evaluate the limit #sqrt(x-1)/(x-4)# as x approaches #3#?

Answer 1

Emily Ammerman

#- sqrt(2)#

We have: #lim_(x -> 3) (sqrt(x - 1)) / (x - 4)#

This limit can be evaluated by substituting #x# with #3#:

#= (sqrt((3) - 1)) / ((3) - 4)#

#= (sqrt(2)) / (- 1)#

#= - sqrt(2)#

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Answer 2

Isaiah Allen

To evaluate the limit of sqrt(x-1)/(x-4) as x approaches 3, we can substitute the value of x into the expression. However, this would result in an undefined expression since the denominator would be zero. To overcome this, we can simplify the expression by multiplying both the numerator and denominator by the conjugate of the numerator, which is sqrt(x-1) + 2. This simplifies the expression to (sqrt(x-1)(sqrt(x-1) + 2))/(x-4). Now, substituting x = 3 into the simplified expression, we get (sqrt(3-1)(sqrt(3-1) + 2))/(3-4), which simplifies to (sqrt(2)(sqrt(2) + 2))/(-1). Further simplification gives (-2(sqrt(2) + 2))/(-1), which equals 2(sqrt(2) + 2). Therefore, the limit of sqrt(x-1)/(x-4) as x approaches 3 is 2(sqrt(2) + 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.