Find the limit (x^2+x-6)/(x+3) when x approach -3.
limit (x^2+x-6)/(x+3) when x approaches -3
first we need to find the value of the function when x=-3
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For evaluating the limit, we'll choose the dividing out technique.
We'll apply the direct substitution, by substituting the unknown x, by the value -3 and we'll see that it fails, because both, numerator and denominator, are cancelling for x=-3. That means x=-3 is a root for both, that means that (x+3) is a common factor for both.
We'll write the numerator using the formula:
x^2+x-6=(x-x1)(x-x2), where x1, x2 are the roots and x1=-3
We also know that x1+x2 = -1, -3+x2=-1
and x1*x2=-6, (-3)*x2=-6
Now, we'll evaluate the limit:
lim (x^2+x-6)/(x+3) = lim (x+3)(x-2)/(x+3)
Now, we can divide out like factor:
lim (x^2+x-6)/(x+3) = lim (x-2)
We can apply the replacement theorem and we'll get:
lim (x-2) = -3-2 = -5
So, lim (x^2+x-6)/(x+3) = -5.