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Math  /  Calculus

Questionlim⁡x→∞(log⁡(1x))\lim _{x \rightarrow \infty}\left(\log \left(\frac{1}{x}\right)\right)

Studdy Solution

STEP 1

1. The function inside the limit is log⁡(1x)\log\left(\frac{1}{x}\right).
2. We are evaluating the behavior of this function as xx approaches infinity.
3. The logarithm function used here is the natural logarithm, log⁡(x)=ln⁡(x)\log(x) = \ln(x).

STEP 2

1. Simplify the expression log⁡(1x)\log\left(\frac{1}{x}\right).
2. Evaluate the limit as xx approaches infinity.

STEP 3

First, simplify the expression log⁡(1x)\log\left(\frac{1}{x}\right) using the logarithmic identity log⁡(1x)=−log⁡(x)\log\left(\frac{1}{x}\right) = -\log(x):
log⁡(1x)=−log⁡(x)\log\left(\frac{1}{x}\right) = -\log(x)

STEP 4

Now, evaluate the limit lim⁡x→∞−log⁡(x)\lim_{x \to \infty} -\log(x). As xx approaches infinity, log⁡(x)\log(x) also approaches infinity. Therefore, −log⁡(x)-\log(x) approaches negative infinity:
lim⁡x→∞−log⁡(x)=−∞\lim_{x \to \infty} -\log(x) = -\infty
The limit is:
−∞\boxed{-\infty}

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