Question
Determine the following IImit- -SHOW WORK in WRITTEN WORKI \( \lim _{x \rightarrow-1}\left(x^{3}-2 x^{2}+x-3\right) \)
Ask by Carter Gordon. in the United States
Mar 19,2025
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Answer
The limit is -7.
Solution
Calculate the limit \( \lim _{x \rightarrow-1}x^{3}-2x^{2}+x-3 \).
Evaluate the limit by following steps:
- step0: Evaluate the limit:
\(\lim _{x\rightarrow -1}\left(x^{3}-2x^{2}+x-3\right)\)
- step1: Rewrite the expression:
\(\lim _{x\rightarrow -1}\left(x^{3}\right)-\lim _{x\rightarrow -1}\left(2x^{2}\right)+\lim _{x\rightarrow -1}\left(x\right)-\lim _{x\rightarrow -1}\left(3\right)\)
- step2: Calculate:
\(-1-2-1-3\)
- step3: Calculate:
\(-3-1-3\)
- step4: Calculate:
\(-4-3\)
- step5: Calculate:
\(-7\)
The limit of \(x^{3}-2x^{2}+x-3\) as \(x\) approaches -1 is -7.
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Simplify this solution Bonus Knowledge
To determine the limit \( \lim _{x \rightarrow -1}\left(x^{3}-2 x^{2}+x-3\right) \), we start by substituting \( x = -1 \) directly into the expression. Now, let's calculate it step by step: 1. Substitute \( x = -1 \): \[ (-1)^{3} - 2(-1)^{2} + (-1) - 3 \] 2. Calculate each term: - \( (-1)^{3} = -1 \) - \( -2(-1)^{2} = -2 \times 1 = -2 \) - \( (-1) = -1 \) - \( -3 = -3 \) 3. Combine these results: \[ -1 - 2 - 1 - 3 = -1 - 2 - 1 - 3 = -4 - 3 = -7 \] Thus, the limit is: \[ \lim_{x \rightarrow -1}(x^{3}-2x^{2}+x-3) = -7 \]
