Graph 1/( natural log of x)

Problem

ƒ(x)=1/ln(x)

Solution

1. Identify the domain of the function. The natural logarithm ln(x) is defined for x>0 Additionally, the denominator cannot be zero, so ln(x)≠0 which means x≠1 The domain is (0,1)∪(1,∞)

2. Determine the vertical asymptotes by finding where the function is undefined. As x→1 the denominator ln(x)→0 resulting in a vertical asymptote at x=1

3. Analyze the end behavior as x approaches the boundaries of the domain. As x→0 ln(x)→−∞ so ƒ(x)→0 As x→∞ ln(x)→∞ so ƒ(x)→0 This indicates a horizontal asymptote at y=0

4. Find the derivative to determine the slope and intervals of increase or decrease. Using the chain rule:

d(ƒ(x))/d(x)=−1/((ln(x))2)⋅1/x

5. Determine the sign of the derivative. Since x>0 and (ln(x))2>0 for all x in the domain, the derivative is always negative. This means the function is strictly decreasing on (0,1) and (1,∞)

6. Sketch the graph based on these features. The curve starts near (0,0) decreases toward −∞ as it approaches x=1 from the left, reappears from +∞ to the right of x=1 and decreases toward the xaxis as x→∞

Final Answer

ƒ(x)=1/ln(x)

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