Pregunta
Determine the following limits \( \lim _{x \rightarrow 0}\left[\frac{4 x^{2}+x-1}{3 x^{2}+2 x+1}\right] \)
Ask by Dickson Wyatt. in South Africa
Mar 13,2025
Solución de inteligencia artificial de Upstudy
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The limit is -1.
Solución
Calculate the limit \( \lim _{x\rightarrow 0} \frac{4x^{2}+x-1}{3x^{2}+2x+1} \).
Evaluate the limit by following steps:
- step0: Evaluate the limit:
\(\lim _{x\rightarrow 0}\left(\frac{4x^{2}+x-1}{3x^{2}+2x+1}\right)\)
- step1: Rewrite the expression:
\(\frac{\lim _{x\rightarrow 0}\left(4x^{2}+x-1\right)}{\lim _{x\rightarrow 0}\left(3x^{2}+2x+1\right)}\)
- step2: Calculate:
\(\frac{-1}{\lim _{x\rightarrow 0}\left(3x^{2}+2x+1\right)}\)
- step3: Calculate:
\(\frac{-1}{1}\)
- step4: Calculate:
\(-1\)
The limit of the expression \( \frac{4x^{2}+x-1}{3x^{2}+2x+1} \) as \( x \) approaches 0 is -1.
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Simplifique esta solución Extra Insights
To determine the limit \[ \lim _{x \rightarrow 0}\left[\frac{4 x^{2}+x-1}{3 x^{2}+2 x+1}\right], \] we can directly substitute \( x = 0 \) into the function: 1. Substitute \( x = 0 \): - The numerator becomes: \[ 4(0)^{2} + 0 - 1 = -1. \] - The denominator becomes: \[ 3(0)^{2} + 2(0) + 1 = 1. \] 2. Now we have: \[ \frac{-1}{1} = -1. \] Therefore, the limit is \[ \boxed{-1}. \]
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