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Score on last try: 0 of 10 pts. See Details for more. \( \rightarrow \) Next question \( \longleftrightarrow \) Get a similar question You can retry this question below Evaluate the limit: \( \lim _{s \rightarrow 5} \frac{s^{3}-125}{s-5}=\square \) Question Help: Video \( \square \) Message instructor Submit Question Jump to Answer
Ask by Hill Deleon. in the United States
Jan 27,2025
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Answer
The limit is 75.
Solution
Calculate the limit \( \lim_{s\rightarrow 5} \frac{s^{3}-125}{s-5} \).
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
\(\lim _{s\rightarrow 5}\left(\frac{s^{3}-125}{s-5}\right)\)
- step1: Reduce the fraction:
\(\lim _{s\rightarrow 5}\left(s^{2}+5s+25\right)\)
- step2: Rewrite the expression:
\(\lim _{s\rightarrow 5}\left(s^{2}\right)+\lim _{s\rightarrow 5}\left(5s\right)+\lim _{s\rightarrow 5}\left(25\right)\)
- step3: Calculate:
\(25+25+25\)
- step4: Calculate:
\(75\)
The limit of \( \frac{s^{3}-125}{s-5} \) as \( s \) approaches 5 is 75.
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Simplify this solution Mind Expander
To evaluate the limit as \( s \) approaches 5 for the expression \( \frac{s^{3}-125}{s-5} \), we need to recognize that \( s^{3} - 125 \) can be factored using the difference of cubes formula, which states \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, \( s^3 - 5^3 \) can be factored as \( (s - 5)(s^2 + 5s + 25) \). So, we can rewrite the limit as: \[ \lim_{s \rightarrow 5} \frac{(s - 5)(s^2 + 5s + 25)}{s - 5} \] Cancelling \( (s - 5) \) gives us: \[ \lim_{s \rightarrow 5} (s^2 + 5s + 25) \] Substituting \( s = 5 \): \[ 5^2 + 5 \cdot 5 + 25 = 25 + 25 + 25 = 75 \] Thus, the limit is \( 75 \).
