Find the Asymptotes f(x) = natural log of x+6

Problem

ƒ(x)=ln(x+6)

Solution

1. Identify the type of function. The function ƒ(x)=ln(x+6) is a logarithmic function. Logarithmic functions of the form y=ln(g(x)) typically have vertical asymptotes but no horizontal or oblique asymptotes.

2. Determine the vertical asymptote by finding where the argument of the natural logarithm is equal to zero. Logarithmic functions are undefined for values less than or equal to zero.

3. Set the argument x+6 equal to zero and solve for x

x+6=0

4. Subtract 6 from both sides to isolate x

x=−6

5. Check for horizontal asymptotes by evaluating the limit of the function as x approaches infinity.

(lim_x→∞)(ln(x+6))=∞

Since the limit is infinite, there is no horizontal asymptote.

Final Answer

Vertical Asymptote: *x=−6

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