How do you determine the limit of #(x^2-2x)/(x^2-4x+4)# as x approaches 2-?

Answer 1

#-oo#

#lim_(x to 2^-) (x^2-2x)/(x^2-4x+4)#

it indeterminate so we can simplify a bit using L'Hopital's Rule

#= lim_(x to 2^-) (2x-2)/(2x-4)#
factor out the 2 #= lim_(x to 2^-) (x-1)/(x-2)#
if we now sub #x = 2 - delta# where #0 < delta " << " 1#, as it is a left sided limit then we have
#= lim_(delta to 0) (2 - delta -1)/(2 - delta-2)#
#= lim_(delta to 0) (1 - delta)/(-delta) = -oo#
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Answer 2

To determine the limit of (x^2-2x)/(x^2-4x+4) as x approaches 2-, we can substitute the value 2 into the expression and simplify. By doing so, we get (2^2-2(2))/(2^2-4(2)+4), which simplifies to (4-4)/(4-8+4). Further simplification gives 0/0. This indicates that the expression is indeterminate at x=2-. To find the limit, we can factor the numerator and denominator. Factoring the numerator gives (x(x-2)), and factoring the denominator gives (x-2)(x-2). Canceling out the common factor (x-2), we are left with x/(x-2). Now, substituting 2 into this simplified expression gives 2/(2-2), which simplifies to 2/0. This is still an indeterminate form. To resolve this, we can use algebraic manipulation or L'Hôpital's rule. Applying L'Hôpital's rule, we differentiate the numerator and denominator separately. The derivative of x is 1, and the derivative of (x-2) is 1. Evaluating the limit of the derivatives as x approaches 2- gives 1/1, which equals 1. Therefore, the limit of (x^2-2x)/(x^2-4x+4) as x approaches 2- is 1.

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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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