lim x-> infinity (sqrt(x+6x^2))/(7x-1)
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.
To find the limit of the given expression as x approaches infinity, we can simplify the expression by dividing both the numerator and denominator by x^2, which is the highest power of x in the expression.
Dividing the numerator sqrt(x+6x^2) by x^2 gives us sqrt(x/x^2 + 6x^2/x^2), which simplifies to sqrt(1/x + 6).
Dividing the denominator 7x-1 by x^2 gives us (7x/x^2 - 1/x^2), which simplifies to 7/x - 1/x^2.
Now, as x approaches infinity, 1/x approaches 0. Therefore, the limit of 1/x^2 as x approaches infinity is also 0.
So, the limit of the expression sqrt(x+6x^2)/(7x-1) as x approaches infinity is the same as the limit of sqrt(1/x + 6)/(7/x - 1/x^2) as x approaches infinity.
Using the limit properties, we can find the limit of the expression by taking the limit of each individual part separately.
The limit of sqrt(1/x + 6) as x approaches infinity is sqrt(0 + 6) = sqrt(6).
The limit of (7/x - 1/x^2) as x approaches infinity is (7/∞ - 1/∞^2) = 0 - 0 = 0.
Therefore, the limit of the original expression is sqrt(6)/0, which is undefined or "does not exist".
To evaluate the limit of the given expression as x approaches infinity, we can simplify the expression by dividing both the numerator and the denominator by the highest power of x.
Let's start by dividing both the numerator and denominator by x^2, which is the highest power of x in the expression.
lim x-> infinity (sqrt(x+6x^2))/(7x-1)
= lim x-> infinity (sqrt(x/x^2 + 6x^2/x^2))/(7x/x^2 - 1/x^2)
Simplifying this expression:
= lim x-> infinity (sqrt(1/x + 6))/(7 - 1/x^2)
Next, we take the limit as x approaches infinity:
= (sqrt(0 + 6))/(7 - 0)
= sqrt(6)/7
Therefore, the limit of the expression as x approaches infinity is sqrt(6)/7.