How do you evaluate each of the following limits, if it exists #lim (3x^2+5x-2)/(x^2-3x-10)# as #x-> -2#?

Answer 1

#lim_(xrarr-2)(3x^2+5x-2)/(x^2-3x-10)=1#

To do this, factor the numerators and denominators. Notice that we can't plug in #-2# now, because this gives us the ratio #0//0#.
To factor #3x^2+5x-2#, we look for two numbers whose product is #-6# and sum is #5#, which are #6# and #-1#.
Then, #3x^2+5x-2=3x^2+6x-x-2#
#color(white)(lll)=3x(x+2)-1(x+2)#
#color(white)(lll)=(3x-1)(x+2)#
In factoring #x^2-3x-10#, we look for the numbers whose product is #-10# and sum is #-3#, which are #-5# and #2#.
Then, #x^2-3x-10=(x-5)(x+2)#
So #lim_(xrarr-2)(3x^2+5x-2)/(x^2-3x-10)=lim_(xrarr-2)((3x-1)(x+2))/((x-5)(x+2))#, and we see that we can cancel the #(x+2)# terms, so the limit equals #lim_(xrarr-2)(3x-1)/(x-5)#.
Now plugging in #-2# won't create the issue it did before and we can plug in #x=-2#, so: #lim_(xrarr-2)(3x-1)/(x-5)=(3(-2)-1)/(-2-5)=(-7)/(-7)=1#.
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Answer 2

To evaluate the limit lim (3x^2+5x-2)/(x^2-3x-10) as x approaches -2, we substitute -2 for x in the expression. This gives us (3(-2)^2+5(-2)-2)/((-2)^2-3(-2)-10). Simplifying further, we have (12-10-2)/(4+6-10). Continuing to simplify, we get 0/0. This is an indeterminate form. To evaluate the limit further, we can use algebraic manipulation or apply L'Hôpital's rule.

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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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