text: How do you find the limit of (n^4+6n^2)/(n^4-2n+3) as n approaches infinity?
  
  Jul 23, 2016
  
  lim_(n->∞) (n^4+6n^2)/(n^4-2n+3) = 1
  
  Explanation:
  
  Since this is also known as an infinite limit, and both the numerator and denominator are polynomials, we can divide each term by the highest power, namely by ${n}^{4}$.
  
  Therefore, we have
  
  lim_(n->∞) (n^4+6n^2)/(n^4-2n+3) = lim_(n->∞) (cancel((n^4/n^4))+(6n^2)/(n^4))/(cancel((n^4)/(n^4))-(2n)/(n^4)+(3)/(n^4))
  
  = lim_(n->∞) (1+6/(n^2))/(1-2/(n^3)+3/(n^4))
  
  Also, since we know that lim_(n->∞) 1/(n^m) = 0 iff m>0, we then have
  
  lim_(n->∞) (1+6/(n^2))/(1-2/(n^3)+3/(n^4)) = 1/(1) = 1
  
  We can also graph our original function to see if this limit makes sense.
  
  graph{(x^4+6x^2)/(x^4-2x+3) [-10, 10, -5, 5]}
  
  In this case, we can see that as n->∞, our function tends to $1$.
explanation: The document contains a math question and a detailed answer explaining how to find the limit of a given function as \( n \) approaches infinity. The answer includes a step-by-step solution and a graph to verify the result.
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distilabel_metadata.content: How do you find the limit of (n^4+6n^2)/(n^4-2n+3) as n approaches infinity?
  
  Jul 23, 2016
  
  lim_(n->∞) (n^4+6n^2)/(n^4-2n+3) = 1
  
  Explanation:
  
  Since this is also known as an infinite limit, and both the numerator and denominator are polynomials, we can divide each term by the highest power, namely by ${n}^{4}$.
  
  Therefore, we have
  
  lim_(n->∞) (n^4+6n^2)/(n^4-2n+3) = lim_(n->∞) (cancel((n^4/n^4))+(6n^2)/(n^4))/(cancel((n^4)/(n^4))-(2n)/(n^4)+(3)/(n^4))
  
  = lim_(n->∞) (1+6/(n^2))/(1-2/(n^3)+3/(n^4))
  
  Also, since we know that lim_(n->∞) 1/(n^m) = 0 iff m>0, we then have
  
  lim_(n->∞) (1+6/(n^2))/(1-2/(n^3)+3/(n^4)) = 1/(1) = 1
  
  We can also graph our original function to see if this limit makes sense.
  
  graph{(x^4+6x^2)/(x^4-2x+3) [-10, 10, -5, 5]}
  
  In this case, we can see that as n->∞, our function tends to $1$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer explaining how to find the limit of a given function as \( n \) approaches infinity. The answer includes a step-by-step solution and a graph to verify the result.
  
  Label: question-and-answer