How do you find all the asymptotes for function #g(x)=e^(x-2)#?

Answer 1

The only asymptote is horizontal, and it is #y=0#, as #x->-infty#.

A function can have horizontal asymptotes (if the limits as #x# approaches #pm infty# exist and are finite), vertical ones (if the domain is not the whole real number set), or oblique ones (if the end behaviour is asymptotically equivalent to a line. Note that, for each of the two directions towards infinity, horizontal and oblique asymptotes are mutually exclusive (actually, horizontal ones are a special case of the oblique ones, since they represent a line with zero slope).

Since the exponential is defined over the whole real set, you can't have vertical asymptotes. So, let's check the infinite limits:

Negative infinity: #lim_{x\to -infty} e^{x-2}#
This limit is very similar to the one of #e^x#, and you can come back to it in two ways:
#lim_{x\to -infty} e^{x-2}=lim_{x\to -infty} e^x / e^2= 1/ e^2lim_{x\to -infty} e^x=1/ e^2 * 0 = 0#
#lim_{x\to -infty} e^{x-2}=lim_{y\to -infty} e^y =0#.
Positive infinity #lim_{x\to infty} e^{x-2}#
Everything we said above, still holds, so you find that the limit is #+infty#, either by algebra (#e^{x-2}=e^x / e^2#), or by substitution (#y=x-2#). So, there is no horizontal asymptote as #x-> infty#. The test to check for oblique ones requires to check the limit of #f(x)/x#. Is that limit is finite, it represents the slope of the line. But I'm assuming that you know that if you divide an exponential by a polynomial (of any degree), the exponential always "wins", and so the limit is still infinite, and there is no oblique asymptote when #x->infty#.
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Answer 2

To find the asymptotes of the function ( g(x) = e^{x-2} ), you need to consider the behavior of the function as ( x ) approaches positive or negative infinity. Since the exponential function ( e^{x} ) grows without bound as ( x ) increases, there are no horizontal asymptotes for ( g(x) ).

However, there is a vertical asymptote at ( x = 2 ), because when ( x ) approaches 2, the denominator ( e^{x-2} ) approaches 0. Therefore, the vertical asymptote for the function ( g(x) ) is ( x = 2 ).

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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.

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.

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.

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.

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