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lim_(x rarr0)(ln(1+x))/(x)=1

lim⁡x→0ln⁡(1+x)x=1 \lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}=1

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Find derivatives of logarithmic functionsright chevron icon

Full solution

Q. lim⁡x→0ln⁡(1+x)x=1 \lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}=1
  1. Recognize the limit form: Recognize the limit form.\newlineWe have lim⁡x→0ln⁡(1+x)x\lim_{x \to 0}\frac{\ln(1+x)}{x}, which looks like a 0/00/0 indeterminate form.
  2. Apply L'Hopital's Rule: Apply L'Hopital's Rule since we have a 0/00/0 form.\newlineTake the derivative of the numerator and the derivative of the denominator separately.
  3. Differentiate the numerator: Differentiate the numerator. ddx(ln⁡(1+x))=11+x\frac{d}{dx}(\ln(1+x)) = \frac{1}{1+x}
  4. Differentiate the denominator: Differentiate the denominator. ddx(x)=1\frac{d}{dx}(x) = 1
  5. Apply the derivatives to the limit: Apply the derivatives to the limit. \lim_{x \to \(0\)}\left(\frac{\(1\)}{\(1\)+x}\right)/\(1
  6. Simplify the expression: Simplify the expression. lim⁡x→011+x=11\lim_{x \to 0}\frac{1}{1+x} = \frac{1}{1}
  7. Evaluate the limit: Evaluate the limit. lim⁡x→011+x=1\lim_{x \to 0}\frac{1}{1+x} = 1

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