Problem to solve
Compute \[ \lim_{ x \to 1 } \frac{x^4+3x^3-13x^2-27x+36}{x^2+3x-4} \]
Solution
STEP 1
Find the following limit: lim_(x->1) (x^4 + 3 x^3 - 13 x^2 - 27 x + 36)/(x^2 + 3 x - 4)
STEP 2
Hint: | Factor the numerator and denominator. (x^4 + 3 x^3 - 13 x^2 - 27 x + 36)/(x^2 + 3 x - 4) = ((x^2 - 9) (x^2 + 3 x - 4))/(x^2 + 3 x - 4): lim_(x->1) ((x^2 - 9) (x^2 + 3 x - 4))/(x^2 + 3 x - 4)
STEP 3
Hint: | Cancel terms. ((x^2 - 9) (x^2 + 3 x - 4))/(x^2 + 3 x - 4) = x^2 - 9: lim_(x->1)(x^2 - 9)
STEP 4
Hint: | The limit of a continuous function at a point is just its value there. lim_(x->1)(x^2 - 9) = 1^2 - 9 = -8: Answer: | | -8
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Edited by tododemate® - 2021