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Find the limit. lim_(x rarr oo)(3x^(4)+2x)/((x^(2)+2)(5x^(2)+1))

11. Find the limit.\newlinelim⁡x→∞3x4+2x(x2+2)(5x2+1) \lim _{x \rightarrow \infty} \frac{3 x^{4}+2 x}{\left(x^{2}+2\right)\left(5 x^{2}+1\right)}

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Find limits of polynomials and rational functionsright chevron icon

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Q. 11. Find the limit.\newlinelim⁡x→∞3x4+2x(x2+2)(5x2+1) \lim _{x \rightarrow \infty} \frac{3 x^{4}+2 x}{\left(x^{2}+2\right)\left(5 x^{2}+1\right)}
  1. Simplify Expression: Simplify the expression by dividing the numerator and the denominator by the highest power of x in the denominator, which is x2x^2.\newlinelim⁡x→∞3x4+2x(x2+2)(5x2+1)=lim⁡x→∞3x2+2x(1+2x2)(5+1x2) \lim_{x \to \infty} \frac{3x^4 + 2x}{(x^2 + 2)(5x^2 + 1)} = \lim_{x \to \infty} \frac{3x^2 + \frac{2}{x}}{(1 + \frac{2}{x^2})(5 + \frac{1}{x^2})}
  2. Divide by Highest Power: As xx approaches infinity, the terms 2x\frac{2}{x} and 2x2\frac{2}{x^2}, 1x2\frac{1}{x^2} approach 00.\newlinelim⁡x→∞3x2+0(1+0)(5+0)=3x25 \lim_{x \to \infty} \frac{3x^2 + 0}{(1 + 0)(5 + 0)} = \frac{3x^2}{5}

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