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Math  /  Calculus

Questionlim⁡x→−4x2+∣x∣−20x+4\lim _{x \rightarrow-4} \frac{x^{2}+|x|-20}{x+4}

Studdy Solution

STEP 1

1. The function is a rational function with a numerator and a denominator.
2. The limit involves a point where the denominator is zero, leading to a potential indeterminate form.
3. We will need to factor the numerator to simplify the expression.

STEP 2

1. Identify the indeterminate form.
2. Factor the numerator.
3. Simplify the expression.
4. Evaluate the limit.

STEP 3

First, substitute x=−4 x = -4 into the expression to check for an indeterminate form:
(−4)2+∣−4∣−20−4+4=16+4−200=00 \frac{(-4)^{2} + |-4| - 20}{-4 + 4} = \frac{16 + 4 - 20}{0} = \frac{0}{0}
This is an indeterminate form.

STEP 4

Factor the numerator x2+∣x∣−20 x^2 + |x| - 20 . Since ∣x∣=x |x| = x for x>0 x > 0 and ∣x∣=−x |x| = -x for x<0 x < 0 , we consider x=−4 x = -4 and use ∣x∣=−x |x| = -x :
x2+∣x∣−20=x2−x−20 x^2 + |x| - 20 = x^2 - x - 20
Now, factor x2−x−20 x^2 - x - 20 :
x2−x−20=(x−5)(x+4) x^2 - x - 20 = (x - 5)(x + 4)

STEP 5

Simplify the expression by canceling the common factor (x+4) (x + 4) in the numerator and the denominator:
(x−5)(x+4)x+4=x−5forx≠−4 \frac{(x - 5)(x + 4)}{x + 4} = x - 5 \quad \text{for} \quad x \neq -4

STEP 6

Evaluate the limit of the simplified expression as x x approaches −4-4:
lim⁡x→−4(x−5)=−4−5=−9 \lim_{x \to -4} (x - 5) = -4 - 5 = -9
The value of the limit is:
−9 \boxed{-9}

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