Gordon Barnett
06/12/2023 · Senior High School
(1 point) Evaluate the limit, using L'Hôpital's Rule. Enter \( \operatorname{INF} \) for \( \infty \), \( \operatorname{\text {INFfor}}-\infty \), or DNE if the limit does not exist, but is neither \( \infty \) nor \( -\infty \). \( \lim _{x \rightarrow 0^{+}} 14 x \ln x= \)
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Step-by-step Solution
Calculate the limit \( \lim_{x\rightarrow 0^{+}} 14x\ln x \).
Evaluate the limit by following steps:
- step0: Evaluate using L'Hopital's rule:
\(\lim _{x\rightarrow 0^{+}}\left(14x\ln{\left(x\right)}\right)\)
- step1: Transform the expression:
\(\lim _{x\rightarrow 0^{+}}\left(\frac{\ln{\left(x\right)}}{\frac{1}{14x}}\right)\)
- step2: Use the L'Hopital's rule:
\(\lim _{x\rightarrow 0^{+}}\left(\frac{\frac{d}{dx}\left(\ln{\left(x\right)}\right)}{\frac{d}{dx}\left(\frac{1}{14x}\right)}\right)\)
- step3: Find the derivative:
\(\lim _{x\rightarrow 0^{+}}\left(\frac{\frac{1}{x}}{-\frac{1}{14x^{2}}}\right)\)
- step4: Simplify the expression:
\(\lim _{x\rightarrow 0^{+}}\left(-14x\right)\)
- step5: Rewrite the expression:
\(-14\times \lim _{x\rightarrow 0^{+}}\left(x\right)\)
- step6: Calculate:
\(-14\times 0\)
- step7: Calculate:
\(0\)
The limit \( \lim _{x \rightarrow 0^{+}} 14 x \ln x \) is equal to 0.
Quick Answer
The limit is 0.
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