Question
\( \varliminf _ { x \rightarrow - 3 } \frac { x ^ { 2 } - 2 x - 15 } { x + 3 } \)
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Feb 24,2025
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Answer
El límite inferior es -8.
Solution
Para evaluar
lim infₓ→₋3 (x² - 2x - 15)⁄(x + 3),
observa que el numerador se puede factorizar:
x² - 2x - 15 = (x - 5)(x + 3).
Por lo tanto, la función se simplifica a
(x - 5)(x + 3)/(x + 3) = x - 5 (provided x ≠ -3).
Aunque en x = -3 la función no está definida, al calcular el límite podemos considerar la función simplificada. Así,
limₓ→₋3 (x - 5) = (-3) - 5 = -8.
Por lo tanto, el límite inferior es -8.
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Simplify this solution Mind Expander
To find \( \varliminf _ { x \rightarrow - 3 } \frac { x ^ { 2 } - 2 x - 15 } { x + 3 } \), we start by simplifying the expression. The numerator \( x^2 - 2x - 15 \) can be factored as \( (x - 5)(x + 3) \). So, we can rewrite the limit: \[ \frac{x^2 - 2x - 15}{x + 3} = \frac{(x - 5)(x + 3)}{x + 3} \] For \( x \neq -3 \), this simplifies to \( x - 5 \). Therefore, we need to find the limit as \( x \) approaches \(-3\): \[ \lim_{x \to -3} (x - 5) = -3 - 5 = -8 \] Since this limit exists and is not affected by the approach (from the left or the right), we have: \[ \varliminf_{x \rightarrow -3} \frac{x^2 - 2x - 15}{x + 3} = -8 \]
