Find the Asymptotes y=1/(x-5)

Problem

y=1/(x−5)

Solution

1. Identify the vertical asymptote by finding the values of x that make the denominator zero while the numerator remains non-zero.

x−5=0

x=5

2. Identify the horizontal asymptote by comparing the degrees of the numerator and the denominator.
   The degree of the numerator is 0 and the degree of the denominator is 1

3. Apply the rule for horizontal asymptotes where the degree of the denominator is greater than the degree of the numerator.

(lim_x→∞)(1/(x−5))=0

(lim_x→−∞)(1/(x−5))=0

y=0

4. Check for oblique asymptotes by determining if the degree of the numerator is exactly one higher than the degree of the denominator.
   Since the degree of the numerator is less than the degree of the denominator, there is no oblique asymptote.

Final Answer

Vertical Asymptote: *x=5, Horizontal Asymptote: *y=0

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