Question
Question 2 Evaluate the following limits: \( 2.1 \lim _{x \rightarrow 3}\left(x^{3}+2\right)\left(x^{2}-5 x\right) \). \( 2.2 \lim _{x \rightarrow-4^{-}}\left(\frac{|x+4|}{x+4}\right) \.
Ask by Schultz Horton. in South Africa
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The limits are:
2.1: -174
2.2: -1
Solution
We are given two limits to evaluate.
2.1: Evaluate limₓ → 3 [(x³ + 2)(x² - 5x)].
Step 1: Factor and substitute x = 3.
• Compute x³ + 2 when x = 3: 3³ + 2 = 27 + 2 = 29.
• Compute x² - 5x when x = 3: 3² - 5(3) = 9 - 15 = -6.
Step 2: Multiply the results: 29 × (-6) = -174.
Therefore, limₓ → 3 [(x³ + 2)(x² - 5x)] = -174.
2.2: Evaluate limₓ → -4⁻ (|x + 4|/(x + 4)).
Step 1: Understand the behavior as x approaches -4 from the left.
Since x → -4⁻, it means x is slightly less than -4, so (x + 4) is negative (but not zero).
Step 2: Use the definition of absolute value. For any negative number u, |u| = -u.
Let u = x + 4 which is negative, then |x + 4| = -(x + 4).
Step 3: Rewrite the fraction:
|x + 4|/(x + 4) = [-(x + 4)]/(x + 4) = -1.
Thus, the limit evaluates to -1.
Final Answers:
2.1: -174
2.2: -1
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Simplify this solution Beyond the Answer
To evaluate the first limit, as \( x \) approaches 3, we can directly substitute 3 into the function. So, \[ \lim _{x \rightarrow 3}\left(x^{3}+2\right)\left(x^{2}-5 x\right) = (3^{3}+2)(3^{2}-5 \cdot 3) = (27 + 2)(9 - 15) = 29 \cdot (-6) = -174. \] For the second limit, we need to consider the behavior of the function as \( x \) approaches -4 from the left. The expression \(\frac{|x+4|}{x+4}\) simplifies based on the sign of \(x + 4\). Since we are approaching from the left, \(x + 4\) is negative, making \(|x + 4| = -(x + 4)\). Thus, \[ \frac{|x+4|}{x+4} = \frac{-(x+4)}{x+4} = -1 \quad \text{as} \quad x \rightarrow -4^{-}. \] So, the limit is \(-1\).
