What are the asymptotes for #f(x) = 3 + log(x+2)#?

Answer 1

For one, the argument of a log-function must be positive.

This means #x> -2# and #x=2# is the corresponding (vertical) asymptote, because you can get as near to 2 as you want. In "the language" we say:

#lim_(x->2+) f(x)= -oo#

There is no horizontal asymptote, as the log function will grow with #x# getting larger, even though the growth is slow, or:

#lim_(x-> oo) f(x) = + oo# graph{3+log(x+2) [-5.67, 14.33, -3.32, 6.68]}

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Answer 2

Ariana Alvarez

The asymptotes for the function ( f(x) = 3 + \log(x+2) ) are as follows:

Vertical asymptote: Since the domain of the logarithmic function is restricted to positive real numbers, the vertical asymptote occurs when the argument of the logarithm, ( x + 2 ), equals zero. Thus, the vertical asymptote is ( x = -2 ).
Horizontal asymptote: Logarithmic functions do not have horizontal asymptotes.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.