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Limit of xln⁡(x)x \ln\left(x\right) as x→0+x \to 0^{+}

The limit of xln⁡(x)x \ln\left(x\right) as x→0+x \to 0^{+} is 00.

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Graph of x * ln(x) near x = 0⁺0.20.40.60.811.21.41.61.82-0.50.511.5limit = 0f(x)
Graph of x * ln(x) near x = 0⁺. The open circle marks the point x = 0⁺, where the function may be undefined; the curve still heads to 0.

How to work it out

  1. 1

    We want what x * ln(x) approaches as x → 0⁺.

    lim⁡x→0+xln⁡(x)\lim_{x \to 0^{+}} x \ln\left(x\right)
  2. 2

    Direct substitution gives an indeterminate form: 0·∞. It does not tell us the limit.

  3. 3

    L'Hôpital's rule: when the form is 0·∞, the limit equals the limit of the ratio of the derivatives.

    lim⁡ln⁡(x)1x=lim⁡1x−1x2\lim \frac{\ln\left(x\right)}{\frac{1}{x}} = \lim \frac{\frac{1}{x}}{\frac{-1}{x^{2}}}
  4. 4

    Still 0·∞ (both vanish again), so apply it once more.

    lim⁡1x−1x2=lim⁡−1x2−((−2)x)(x2)2\lim \frac{\frac{1}{x}}{\frac{-1}{x^{2}}} = \lim \frac{\frac{-1}{x^{2}}}{\frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}}
  5. 5

    Still 0·∞ (both vanish again), so apply it once more.

    lim⁡−1x2−((−2)x)(x2)2=lim⁡−((−2)x)(x2)22(x2)2−−((−2)x2x22x)((x2)2)2\lim \frac{\frac{-1}{x^{2}}}{\frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}} = \lim \frac{\frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}}{\frac{2 \left(x^{2}\right)^{2} - -\left(\left(-2\right) x 2 x^{2} 2 x\right)}{\left(\left(x^{2}\right)^{2}\right)^{2}}}
  6. 6

    Method: rewrite as ln x / (1/x) and use L'Hôpital: the logarithm grows slower than any power.

  7. 7

    Result:

    lim⁡x→0+xln⁡(x)=0\lim_{x \to 0^{+}} x \ln\left(x\right) = 0
LaTeX source of the formulas on this page
\[ \lim_{x \to 0^{+}} x \ln\left(x\right) = 0 \]
\[ \lim_{x \to 0^{+}} x \ln\left(x\right) \]
\[ \lim \frac{\ln\left(x\right)}{\frac{1}{x}} = \lim \frac{\frac{1}{x}}{\frac{-1}{x^{2}}} \]
\[ \lim \frac{\frac{1}{x}}{\frac{-1}{x^{2}}} = \lim \frac{\frac{-1}{x^{2}}}{\frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}} \]
\[ \lim \frac{\frac{-1}{x^{2}}}{\frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}} = \lim \frac{\frac{-\left(\left(-2\right) x\right)}{\left(x^{2}\right)^{2}}}{\frac{2 \left(x^{2}\right)^{2} - -\left(\left(-2\right) x 2 x^{2} 2 x\right)}{\left(\left(x^{2}\right)^{2}\right)^{2}}} \]

Values of x * ln(x) as x → 0⁺ (computed)

xf(x)Distance from the limit
0.1-0.230258512.30e-1
0.01-0.0460517024.61e-2
0.001-0.00690775536.91e-3
0.0001-0.000921034049.21e-4
0.00001-0.000115129251.15e-4

Questions

What is the limit of x * ln(x) as x approaches 0⁺?

The limit of xln⁡(x)x \ln\left(x\right) as x→0+x \to 0^{+} is 00.

Which form is it?

0·∞.

More well-known limits

    Limit of x * ln(x) as x → 0⁺: 0 | WordGPT