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Does f(x) = x - ln(x) have an oblique asymptote?

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That’s not what an oblique asymptote is. An asymptote is oblique if the derivative approaches a non-zero constant, because the difference between it and the asymptote line approaches zero, but the limit of just the function is an infinity. This doesn’t guarantee it has an actual asymptote, because you still have to find it.

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