What is the limit as x approaches infinity of ln(3x^2+2)-ln(x^2) ?

The limit as x approaches infinity of ln(3x^2+2)-ln(x^2) is ln(3)

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limit as x approaches infinity of ln(3x^2+2)-ln(x^2)

x \to \infty lim (ln (3 x^{2} + 2) - ln (x^{2}))

ln (3)

Decimal

1.09861 \dots

Solution steps

x \to \infty lim (ln (3 x^{2} + 2) - ln (x^{2}))

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (3 x^{2} + 2) - ln (x^{2}) = ln (\frac{3 x ^{2} + 2}{x ^{2}})

= x \to \infty lim (ln (\frac{3 x ^{2} + 2}{x ^{2}}))

Apply the Limit Chain Rule:

ln (3)

= ln (3)

x \to \infty lim ([x sin (\frac{1}{x})])

x \to (0) lim (\frac{1 - 1 + x y}{x y})

x \to 0 lim (+ \frac{1}{x})

x \to 3 lim (\frac{3 - 12 - x}{x - 3})

x \to 0 lim (\frac{1}{cos ( 6 x )})

What is the limit as x approaches infinity of ln(3x^2+2)-ln(x^2) ?
The limit as x approaches infinity of ln(3x^2+2)-ln(x^2) is ln(3)