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Lim (2x-1)2-9 / x+1 x->-1 How to Evaluate the limit ?

Answer 1

Ezekiel Alexander

#-12#

#lim_(x->-1)((2x-1)^2-9)/(x+1)#

When we try to evaluate this directly we get:

#lim_(x->-1)((2x-1)^2-9)/(x+1)#

#=((2(-1)-1)^2-9)/((-1)+1)=((-3)^2-9)/(1-1)->0/0#

So the limit is indeterminate. A route to evaluating is by first expanding the bracket then simplifying the fraction like so:

#lim_(x->-1)((2x-1)^2-9)/(x+1)#

#=lim_(x->-1)(4x^2-4x+1-9)/(x+1)#

#=lim_(x->-1)(4x^2-4x-8)/(x+1)#

#=lim_(x->-1)(4(x^2-x-2))/(x+1)#

#=lim_(x->-1)(4(x+1)(x-2))/(x+1)#

#=lim_(x->-1)(4cancel((x+1))(x-2))/cancel(x+1)#

#=lim_(x->-1)4(x-2)=-12#

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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.

Lim (2x-1)2-9 / x+1 x-&gt;-1 How to Evaluate the limit ?

Lim (2x-1)2-9 / x+1 x->-1 How to Evaluate the limit ?