Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). lim_(x rarr oo)(sqrt(64x^(6)+12x^(5)+42x^(4)))/(7+6x^(2)) Answer:

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Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).  lim_(x rarr oo)(sqrt(64x^(6)+12x^(5)+42x^(4)))/(7+6x^(2)) Answer:

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Factor Out Highest Power: We are given the limit expression lim_(x -> ∞)(sqrt(64x^(6)+12x^(5)+42x^(4)))/(7+6x^(2)). To simplify this, we will first factor out the highest power of x in the numerator inside the square root.Factor Out x^6: Factor out x^6 from the square root in the numerator to simplify the expression inside the square root. sqrt(64x^(6)+12x^(5)+42x^(4)) = sqrt(x^(6)(64+12/x+42/x^(2)))Take x^3 Out: Now, we can take x^3 out of the square root, since sqrt(x^(6)) = x^(3). sqrt(x^(6)(64+12/x+42/x^(2))) = x^(3)sqrt(64+12/x+42/x^(2))Divide by x^2: Next, we will divide both the numerator and the denominator by x^2, the highest power of x in the denominator. (x^(3)sqrt(64+12/x+42/x^(2)))/(7+6x^(2)) = (x^(3)/x^(2))(sqrt(64+12/x+42/x^(2)))/(7/x^(2)+6)Simplify Expression: Simplify the expression by canceling out x^2 from x^(3)/x^(2) and simplifying the square root as x approaches infinity. (x^(3)/x^(2))(sqrt(64+12/x+42/x^(2)))/(7/x^(2)+6) = x(sqrt(64+12/x+42/x^(2)))/(7/x^(2)+6)Approach Infinity: As x approaches infinity, the terms 12/x, 42/x^(2), and 7/x^(2) in the expression will approach 0. x(sqrt(64+12/x+42/x^(2)))/(7/x^(2)+6) = x(sqrt(64+0+0))/(0+6)Find Limit: Now we have a simplified expression where we can easily find the limit as x approaches infinity. x(sqrt(64))/(6) = x(8)/(6) = (4/3)xThe limit of (4/3)x as x approaches infinity is infinity, which means the limit does not exist (DNE). lim_(x -> ∞)(4/3)x = ∞

Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt{64 x^{6}+12 x^{5}+42 x^{4}}}{7+6 x^{2}}$ $\newline$ Answer:

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Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt{64 x^{6}+12 x^{5}+42 x^{4}}}{7+6 x^{2}}$ $\newline$ Answer: