John J.

asked • 07/30/17

Calculus Question

Find the horizontal asymptote of f(x) = 2^-x+3.
 
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Paul F. answered • 07/30/17

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But what about  x => - ∞ ?
f(x)  =>  ∞
This is typical of an exponential function
Check f(x) = e(-x)   OR  f(x) = e(x)
In one direction f(x) approaches a value but in the
other direction it approaches ∞
So the answer  y = 3  can not be right.
 
Horizontal asymptotes usually are found for Rational Expressions like  (x - 3) / (x +2)
not Exponentials.
 
And this is a Pre-Calculus question, not Calculus.
 
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Michael J.

So (1 / 2x) is not a rational expression? The function has a numerator of 1 and denominator of 2x.  The +3 is just a vertical shift.
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07/31/17

Michael J. answered • 07/30/17

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f(x) = (1/2x) + 3
 
 
If we evaluate f(x) at x=infinity, the value of f(x) becomes 3.  Therefore, the horizontal asymptote is
 
y=3
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Paul F.

But what about  x => - ∞ ?
f(x)  =>  ∞
This is typical of an exponential function
Check f(x) = e(-x)   OR  f(x) = e(x)
In one direction f(x) approaches a value but in the
other direction it approaches ∞
So the answer  y = 3  can not be right.
 
Horizontal asymptotes usually are found for Rational Expressions like  (x - 3) / (x +2)
not Exponentials. 
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07/30/17

Arturo O.

As x→∞, 2-x+3 → 3, so the function converges toward 3.  But as x→-∞, the function diverges toward ∞.  It is valid to say y=3 is a horizontal asymptote, per Michael's answer.    
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07/30/17

Arturo O.

The example of (x-3)/(x+2) would produce a vertical asymptote at x=-2, not a horizontal asymptote.
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07/30/17

Paul F.

Arturo
With all due respect
y = (x-3) / (x+2)  does indeed have a horizontal asymptote; y = 1
And, yes, y=3 would be a horizontal asymptote for y = 2-x + 3  if qualified as  y=3, x > 0
 
Michael J
Rational expressions are usually thought to be more than something with a denominator.
They are typically ratios of polynomials or other algebraic expressions [trig, log or exp].
In my years of teaching, Analytic Geometry covers graphing Rational Expressions
which in most curricula are polynomials, usually binomials or trinomials. 
But this is not written in stone so 2-x could be considered a ratio too.
 
 
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07/31/17

Arturo O.

OK, y = (x-3)/(x+2) has a horizontal asymptote of y=1 and a vertical asymptote of x=-2.
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07/31/17

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