Question
1. Calcule el siguiente limite \( \lim _{x \rightarrow 2} \frac{(x+3)^{2}-3}{x-3} \)
Ask by Hodgson Ingram. in Mexico
Mar 18,2025
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Tutor-Verified Answer
Answer
El límite es -22.
Solution
Calculate the limit \( \lim_{x\rightarrow 2} \frac{(x+3)^{2}-3}{x-3} \).
Evaluate the limit by following steps:
- step0: Evaluate the limit:
\(\lim _{x\rightarrow 2}\left(\frac{\left(x+3\right)^{2}-3}{x-3}\right)\)
- step1: Subtract the terms:
\(\lim _{x\rightarrow 2}\left(\frac{x^{2}+6x+6}{x-3}\right)\)
- step2: Rewrite the expression:
\(\frac{\lim _{x\rightarrow 2}\left(x^{2}+6x+6\right)}{\lim _{x\rightarrow 2}\left(x-3\right)}\)
- step3: Calculate:
\(\frac{22}{\lim _{x\rightarrow 2}\left(x-3\right)}\)
- step4: Calculate:
\(\frac{22}{-1}\)
- step5: Calculate:
\(-22\)
El límite de la expresión dada es -22.
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Simplify this solution Beyond the Answer
Para calcular el límite \( \lim _{x \rightarrow 2} \frac{(x+3)^{2}-3}{x-3} \), primero evaluamos el numerador: \[ (x+3)^{2} - 3 = (2 + 3)^{2} - 3 = 5^{2} - 3 = 25 - 3 = 22 \] El denominador al evaluar en \( x = 2 \) es \( 2 - 3 = -1 \). Así que el límite se simplifica a: \[ \lim_{x \rightarrow 2} \frac{22}{-1} = -22 \] Por lo tanto, el límite es \( -22 \).
