Question
lim x-> infinity (sqrt(x+6x^2))/(7x-1)
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GPT 3.5
To evaluate the limit as x approaches infinity of (sqrt(x+6x^2))/(7x-1), we can simplify the expression by taking the highest power of x in both the numerator and denominator.
Since x is approaching infinity, the term with the highest power of x will dominate the expression. In this case, it is 6x^2 in the numerator and 7x in the denominator.
Dividing both the numerator and denominator by x^2, we get:
lim x-> infinity (sqrt(1/x+6))/(7-1/x)
As x approaches infinity, 1/x approaches 0. Therefore, we have:
lim x-> infinity (sqrt(1/x+6))/(7-1/x)
= lim x-> infinity (sqrt(0+6))/(7-0)
= sqrt(6)/7
Thus, the limit as x approaches infinity of (sqrt(x+6x^2))/(7x-1) is sqrt(6)/7.
Since x is approaching infinity, the term with the highest power of x will dominate the expression. In this case, it is 6x^2 in the numerator and 7x in the denominator.
Dividing both the numerator and denominator by x^2, we get:
lim x-> infinity (sqrt(1/x+6))/(7-1/x)
As x approaches infinity, 1/x approaches 0. Therefore, we have:
lim x-> infinity (sqrt(1/x+6))/(7-1/x)
= lim x-> infinity (sqrt(0+6))/(7-0)
= sqrt(6)/7
Thus, the limit as x approaches infinity of (sqrt(x+6x^2))/(7x-1) is sqrt(6)/7.
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