Infinite Limit Shortcut!! (Calculus)

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[EvTheFlickFan]

I can help! Here's a guide:

For our purposes, n will be the degree of the highest degree term in the numerator, and similarly for m with the denominator.

Ex: ( 2x^3 ) / ( x^2 ); n = 3, m = 2.

IF n > m...

...then the sign of the limit is affected by whether x approaches positive or negative infinity. To determine the sign, check the coefficient of the term of degree n.

Ex: lim {x-> +inf} ( 2x^2 )/x = +infinity
Ex: lim {x-> -inf} ( 2x^2 )/x = -infinity
Ex: lim {x -> +inf} ( -2x^2 )/x = -infinity
Ex: lim {x -> -inf} ( -2x^2 )/x = +infinity

See the pattern? Both the sign of the infinity and the sign of the coefficient must be considered.

+infinity, +coefficient -> +infinity
-infinity, +coefficient -> -infinity
+infinity, -coefficient -> -infinity
-infinity, -coefficient -> +infinity



IF n < m...

...then the limit is zero.

Ex: lim {x-> +inf} x/(x^2) = 0
Ex: lim {x-> +inf} 0/(x^2) = 0**
Ex: lim {x-> -inf} x/(x^2) = 0*

*Notice how the sign of the infinity has no affect. Because, in practice, it just generates -0, which is equal to 0.

**Now, I must say that because the actual function here, f(x) = 0/x^2, can't be defined for x = 0, as f(0) = 0/0 and therefore doesnt exist. So the limit of the function does exist at x = 0, but yhe function itself does not. For the curious, this means f(x) is not continuous at x = 0. We can then infer the definition of continuity is lim{n -> x} f(x) = f(n) where n is any x in the domain of f(x). This is a calculus definition, so if you are in an algebra or precalculus class, you won't need to know this yet.



IF n = m...

...then the limit equals a/b. The variable a here refers the coefficient of the term with degree n, and b the coefficient of the term with degree m.

Ex: ( 2x^2 ) / ( 5x^ 2 ); a = 2, b = 5

Due note that when n = m, the sign of infinity does not affect the sign of the limit.

Ex: lim {x -> +inf} ( 2x^2 )/( 5x^ 2 ) = 2/5
Ex: lim {x -> -inf} ( 2x^2 )/( 5x^ 2 ) = 2/5

However, the signs of a and b do affect the sign of the limit.

Ex: lim {x -> +inf} ( -2x^2 )/( 5x^ 2 ) = -2/5

[PrasangaTimalsena]

its just L'Hospital's rule. It would be better if someone teached that instead of saying its a shortcut or trick

[benthepen]

What if there's an equal exponent on the top and bottom?

[vinnology]

Why do i see this the day after my chapter test

[saikorimilli]

what would the answer be if it was limit x goes to negative infinity?

[Enomaru]

I could've used this about 5 minutes ago.