Math  /  Calculus

Question21 Mark for Review
The function mm is given bs m(x)=log10e+log10(x1)m(x)=\log _{10} e+\log _{10}\left(x^{-1}\right). Which of the following statements about mm is true? (A) mm is increasing, the graph of mm is concave up, and limxm(x)=log10e\lim _{x \rightarrow-\infty} m(x)=\log _{10} e. (B) mm is increasing, the graph of mm is concave down, and limx0+m(x)=\lim _{x \rightarrow 0^{+}} m(x)=-\infty. (C) mm is decreasing, the graph of mm is concave up, and limx0+m(x)=\lim _{x \rightarrow 0^{+}} m(x)=\infty. (D) mm is decreasing, the graph of mm is concave down, and limxm(x)=log10e\lim _{x \rightarrow-\infty} m(x)=-\log _{10} e

Studdy Solution
Evaluate the limit of m(x) m(x) as x x \to -\infty .
Since x x cannot be negative for the logarithm to be defined, this limit is not applicable. Instead, we consider the behavior as x x \to \infty .
As x x \to \infty , ex0\frac{e}{x} \to 0, so log10(ex)\log_{10}\left(\frac{e}{x}\right) \to -\infty.
Therefore, limxm(x)=\lim_{x \to \infty} m(x) = -\infty.
The correct statement about m m is: (C) m m is decreasing, the graph of m m is concave down, and limx0+m(x)=\lim_{x \to 0^+} m(x) = \infty.

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