Find the Asymptotes f(x) = natural log of x-4

Problem

ƒ(x)=ln(x−4)

Solution

1. Identify the domain of the function. The natural logarithm ln(u) is only defined for u>0

2. Set up the inequality for the argument of the logarithm.

x−4>0

3. Solve for x to find the vertical boundary.

x>4

4. Determine the vertical asymptote. As x approaches 4 from the right, the argument x−4 approaches 0 causing the function to approach negative infinity.

(lim_x→4)(ln(x−4))=−∞

5. Check for horizontal asymptotes. As x approaches infinity, the natural logarithm function grows without bound.

(lim_x→∞)(ln(x−4))=∞

6. Conclude that there are no horizontal asymptotes because the limit does not approach a finite constant.

Final Answer

Vertical Asymptote: *x=4

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