What is the limit as x approaches 0+of xln(x+x^2) ?

The limit as x approaches 0+of xln(x+x^2) is 0

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limit as x approaches 0+of xln(x+x^2)

x \to 0 + lim (x ln (x + x^{2}))

0

Solution steps

x \to 0 + lim (x ln (x + x^{2}))

Rewrite for L'Hopital

= x \to 0 + lim (\frac{ln ( x + x ^{2} )}{\frac{1}{x}})

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{\frac{1 + 2 x}{x + x ^{2}}}{- \frac{1}{x ^{2}}})

= x \to 0 + lim (\frac{\frac{1 + 2 x}{x + x ^{2}}}{- \frac{1}{x ^{2}}})

Simplify

\frac{\frac{1 + 2 x}{x + x ^{2}}}{- \frac{1}{x ^{2}}} : - \frac{x ( 2 x + 1 )}{1 + x}

= x \to 0 + lim (- \frac{x ( 2 x + 1 )}{1 + x})

Plug in the value

x = 0

= - \frac{0 \cdot ( 2 \cdot 0 + 1 )}{1 + 0}

Simplify

- \frac{0 \cdot ( 2 \cdot 0 + 1 )}{1 + 0} : 0

= 0

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What is the limit as x approaches 0+of xln(x+x^2) ?
The limit as x approaches 0+of xln(x+x^2) is 0