Pregunta
3) \( \lim _{x \rightarrow-5} \frac{x^{2}-25}{(x+5)(x-6)} \)
Ask by Wilson Simpson. in the United States
Feb 18,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The limit is \( \frac{10}{11} \) or approximately 0.90909.
Solución
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.
Respondido por UpStudy AI y revisado por un tutor profesional
Como
error msg
Explicar 
Simplifique esta solución The Deep Dive
To evaluate the limit \( \lim _{x \rightarrow-5} \frac{x^{2}-25}{(x+5)(x-6)} \), first notice that the numerator can be factored using the difference of squares: \[ x^2 - 25 = (x - 5)(x + 5) \] Now substitute this into the limit expression: \[ \lim _{x \rightarrow -5} \frac{(x - 5)(x + 5)}{(x + 5)(x - 6)} \] We can cancel the \( (x + 5) \) terms (noting that \( x \neq -5 \) when evaluating the limit): \[ \lim _{x \rightarrow -5} \frac{x - 5}{x - 6} \] Now we can safely substitute \( x = -5 \): \[ \frac{-5 - 5}{-5 - 6} = \frac{-10}{-11} = \frac{10}{11} \] Thus, the limit is: \[ \lim _{x \rightarrow -5} \frac{x^{2}-25}{(x+5)(x-6)} = \frac{10}{11} \]
preguntas relacionadas
¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde
Hazte Premium
