Question
3) \( \lim _{x \rightarrow-5} \frac{x^{2}-25}{(x+5)(x-6)} \)
Ask by Lynch Peterson. in Saudi Arabia
Feb 20,2025
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Tutor-Verified Answer
Answer
The limit is \( \frac{10}{11} \) or approximately 0.90909.
Solution
Calculate the limit \( \lim_{x \rightarrow -5} \frac{x^{2}-25}{(x+5)(x-6)} \).
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
\(\lim _{x\rightarrow -5}\left(\frac{x^{2}-25}{\left(x+5\right)\left(x-6\right)}\right)\)
- step1: Reduce the fraction:
\(\lim _{x\rightarrow -5}\left(\frac{x-5}{x-6}\right)\)
- step2: Rewrite the expression:
\(\frac{\lim _{x\rightarrow -5}\left(x-5\right)}{\lim _{x\rightarrow -5}\left(x-6\right)}\)
- step3: Evaluate:
\(\frac{-10}{\lim _{x\rightarrow -5}\left(x-6\right)}\)
- step4: Evaluate:
\(\frac{-10}{-11}\)
- step5: Reduce the fraction:
\(\frac{10}{11}\)
The limit of the expression \( \frac{x^{2}-25}{(x+5)(x-6)} \) as \( x \) approaches -5 is \( \frac{10}{11} \) or approximately 0.90909.
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Simplify this solution Bonus Knowledge
To solve the limit \( \lim _{x \rightarrow -5} \frac{x^{2}-25}{(x+5)(x-6)} \), we can start by simplifying the expression. Notice that \( x^{2} - 25 \) can be factored as \( (x-5)(x+5) \). Thus, we have: \[ \frac{x^{2}-25}{(x+5)(x-6)} = \frac{(x-5)(x+5)}{(x+5)(x-6)} \] Now, we can cancel the \( x+5 \) term (as long as \( x \neq -5 \)): \[ \frac{x-5}{x-6} \] Now we can safely evaluate the limit as \( x \) approaches -5: \[ \lim_{x \rightarrow -5} \frac{x-5}{x-6} = \frac{-5-5}{-5-6} = \frac{-10}{-11} = \frac{10}{11} \] So, the answer is \( \frac{10}{11} \).
