limits at plus and minus infinity and horizontal asmp tootes limits in plus and minus infinity if F ofx becomes arbitrarily close to a number P for all sufficiently large positive X we say that P is the limit of f ofx as X approaches infinity and we write limit as X approaches Infinity of f ofx = p if F ofx becomes arbitrarily close to a numbered Q for all sufficiently large Negative X we say that Q is the limit of f ofx as X approaches negative Infinity which we write as the limit as X approaches minus infinity of F ofx = Q to illustrate these ideas let's consider the function whose graph is shown here the middle portion of the graph is omitted because it's not relevant to either the limit as X goes to Infinity or the limit as X goes to minus infinity the number P indicated here is the limit of f ofx as X goes to Infinity notice that an open interval centered at P corresponds to a horizontal strip in which the graph lies for all sufficiently large positive X this remains true as the width of that open interval and thus the width of the strip decreases all of this essentially means that the function is approxim L constant for large positive X in other words the graph approaches a horizontal line which we call a horizontal ASM toote in this case a rightward horizontal ASM toote the number Q indicated here is the limit of f ofx as X approaches negative Infinity notice that an open interval centered at Q corresponds to a horizontal strip in which the graph lies for all sufficiently large Negative X and this remains true as the width of that open interval decreases so the function is approximately constant for large negx and the graph has a leftward horizontal ASM toote in summary if the limit as X approaches Infinity of f ofx equal p then the line yal p is a rightward horizontal ASM toote if the limit as X approaches negative Infinity of F ofx = Q then the line y = q is a left W horizontal ASM toote examples let's consider first the function f ofx = x + 1 /x here's the graph of that function and notice that the line Y equals 1 is both a rightward and a leftward horizontal ASM toote so the limit as X goes to Infinity of f ofx = 1 and the limit as X goes to minus infinity of f ofx = 1 to understand a little bit better why both of those limits are one notice that the function can be Rewritten as 1 + 1/x when X is large whether positive or negative 1 /x is small and so the value of the function is approximately equal to one we point out also that there's a vertical ASM toote at x equals 0 because of the fact that the function has infinite one-sided limits as X approaches zero from the left and the right another example FX = 1 /x^2 + 1 here's the graph of that function notice that y equals 0 is a horizontal ASM toote both rightward and leftward and so the limit as X approaches plus infinity is zero and the limit as X approaches minus infinity is also zero here's a bit more complicated rational function FX = 2xb - x^2 - 2 / X Cub - 1 here's the graph of that function and here the line yal 2 is both a rightward and a leftward horizontal ASM toote so the limit as X goes to Infinity of f ofx = 2 and the limit as X approaches minus infinity of f ofx equal 2 now to get a sense of why exactly those limits turn out to be two let's notice that the function can be Rewritten as 2 - x^2 /x Cub -1 either by long division or by separating the fraction into two terms appropriately since the second term is a proper fraction that is the degree of the denominator is larger than the degree of the numerator that fraction will be small when X is large since the denominator will be much larger than the numerator and so for large positive or negative x f X will be approximately equal to 2 now notice also that we have a vertical ASM toote at x equal to one because that's where the denominator is equal to zero and the one-sided limits as X approaches one are both infinite here's another example involving a rational function here's the graph of the function the line yal 3 is both a rightward and a leftward horizontal ASM toote so the limit as X goes to Infinity of f ofx = 3 and the limit as X goes to minus infinity of f ofx is also three and notice that the function can be Rewritten as 3 plus the proper fraction 3x /x^2 + 1 now all of the examples we've shown so far have corresponding rightward and leftward horizontal ASM tootes that is the limit as X approaches plus infinity and the limit as X approaches minus infinity coincide now here's an example where that's not the case let F ofx equal x / the sare < TK of x^2 + 1 here's the graph of that function notice that yal 1 is a rightward horizontal ASM toote while yal minus1 is a leftward horizontal ASM toote the limit as X approaches Infinity of f ofx is equal to posi 1 while the limit as X approaches negative Infinity of f ofx is equal to -1 now let's consider F ofx = sin of 2x - x over the < TK x^2 + 1 here's the graph of that function and here y = 1 is the leftward horizontal ASM toote While y = minus1 is a rightward horizontal ASM toote notice how the graph oscillates about each of these horizontal ASM tootes the limit as X approaches Infinity here is minus1 while the limit as X approaches minus infinity of f ofx is + one now let's consider the simple cubic polinomial 6 + 7 x - x cubed here's the graph of that function the graph has no horizontal Asm toote but still we can use the language of limits at plus and minus infinity to describe the behavior of the function for large values of X here we say that the limit as X approaches minus infinity of f ofx is plus infinity while the limit as X approach is positive Infinity of f ofx is minus infinity these statements simply describe the direction in which the tals of the graph go let f ofx x = x * 1 + x - the absolute value of x all / the < TK of x^2 + 1 here the graph has a rightward horizontal ASM toote yal 1 but it does not have a leftward horizontal ASM toote notice here that for negative values of X the numerator of that fraction is equal to x * 1 + 2x while for positive X the numerator is simply equal to X here the limit as X approaches minus infinity of f ofx is plus infinity while the limit as X approaches Infinity of f ofx is equal to 1 for an example in which limits at plus and minus infinity not only don't exist but cannot be described as either plus or minus infinity let's consider F ofx = cine X here the function is bounded between -1 and 1 and oscillates between minus1 and one as X gets larger in either direction and so we can only say that the limit as X goes to Infinity of f ofx does not exist and the limit as X approaches minus infinity of f ofx does not exist here's another example in which neither of the limits at plus and minus infinity exists f ofx = x cosine X here the limit as X goes to Infinity if F ofx does not exist and the same is true for the limit as X goes to minus infinity calculation of limits at plus and minus infinity some basic limits for any P greater than zero the limit as X goes to Infinity of 1/ x p is equal to zero and if x to the p is defined when X is negative the limit as X goes to minus infinity of 1/x the p is equal to 0 Let's look at an example to see how these limits are used let F ofx equal the rational function 3x Cub + x^2 - 5 over 2xb - x + 1 let's find the limit as X goes to infinity and the limit as X goes to minus infinity of f ofx we'll first multiply the numerator and denominator of our fraction by 1 /x Cub the numerator then becomes 3 + 1 /x - 5x cubed ided by 2 - 1x^2 + 1X cubed now we notice that each of the reciprocal powers of X in our expression approach zero as X approaches infinity or minus infinity consequently the limit as X goes to either plus or minus infinity of f ofx is equal to 3es here's another example let F ofx equal x^2 + x + 3 over 5x Cub - 1 notice here that the degree of the denominator is now larger than the degree of the numerator here we'll again multiply the numerator and denominator by one over the highest power of X in the denominator 1/x cubed the numerator now becomes 1 /x + 1x^2 + 3x cubed and the denominator becomes 5 - 1X Cub now again each each of these reciprocal powers of X approaches zero as X goes to plus or minus infinity and so F ofx approaches 0 over five or 0er as X goes to plus or minus infinity another example let f ofx = x Cub - x^2 + 3 over 3x^2 + x - 5 notice here that the degree of the numerator is now greater than the degree of the denominator and let's find the limit as X goes to plus and minus infinity of f ofx we'll take the same approach as before we'll multiply the numerator and denominator by one over the highest power of X in the denominator this time the numerator becomes x + 1 + 3x^2 the denominator becomes 3 + 1 /x - 5x^2 again the reciprocal powers of X go to zero leaving x + 1 over 3 which simply becomes larger and larger as X gets larger and larger and so our limit is plus infinity as X goes to plus infinity and minus infinity as X goes to minus infinity now let's think about rational functions in general suppose that f f of x is a rational function whose numerator is an MTH degree polinomial with coefficients a0 A1 through a m and whose denominator is an Nth Degree polinomial with coefficients B 0 B1 through BN we'll multiply the numerator and denominator by 1 /x the nth power and obtain an expression in which the first term in the denominator is the constant BN notice that all other terms in the denominator involve negative powers of X which will approach zero as X goes to either plus or minus infinity so those negative Powers along with whatever negative Powers appear in the numerator approach zero as X goes to infinity and the value of the function will be approximately equal to the leading term in the numerator divided by BN for large values of X how this behaves for large values of X depends upon the exponent M minus n if m is less than n then the numerator involves a negative power of X which would approach zero as X goes to either plus or minus infinity if m is equal to n then that fraction is simply a subm / B subm which will be the limit as X goes to either plus or minus infinity if m is greater than n then we have a positive power of X in the numerator and so the limit will either be plus infinity or minus infinity so we have three cases when the degree of the denominator is greater than the degree of the numerator the limit as X goes to either plus or minus infinity will be zero if the degrees of the numerator and the denominator are the same then the limit as X goes to either plus or minus infinity will be the ratio of the leading coefficients if the degree of the numerator is greater than the degree of the denominator then the limit doesn't exist and the function will approach either plus infinity or minus INF Infinity as X goes to either plus infinity or minus infinity now let's note that it's not necessary that M and N be integers here for example the limit as X goes to Infinity of 5 x to 4/3 power - x 1/2 power over 3x 4/3 power - 7 x + x 2/3 power will equal 5/3 that is the ratio of the co coefficients of the highest powers in the numerator and denominator since those powers are the same now let's look at a few examples involving functions that are not rational functions for instance let's find the limit as X approaches infinity and the limit as X approaches minus infinity of the function f ofx = 3x minus twice the absolute value of x / x + 2 the first thing we want to notice is that if x is positive absolute value of x is the same as X and so the numerator of the fraction is identical to 3x - 2x or just x if x is negative then the absolute value of x is equal to Min - x and so the numerator of the fraction equal 3x + 2x = 5x now let's notice that each part is a rational function whose numerator and denominator have the same degree 3 so the limit as X goes to plus infinity is equal to the Limit as X goes to Infinity of x/ X +2 which will equal one the ratio of the leading coefficients as X goes to negative Infinity the limit of f of x is the same as the limit of 5x / x + 2 which will be 5 / 1 = 5 here's another example find the limit as X goes to infinity and the limit as X goes to minus infinity for the function f ofx = 3x - 7 / the square < TK of 5x^2 + 4 let's first Factor an X sar from the two terms inside the radical in the denominator 5x^2 + 4 is = to x^2 * 5 + 4 / x^2 now let's rewrite the denominator as the < TK of x^2 * the < TK 5 + 4x^2 and rewrite the < TK of x^2 as the absolute value of x now let's multiply the numerator and denominator of this fraction by 1 /x for positive X the absolute value of x * 1 /x will equal 1 and so for positive X the function becomes 3 - 7 /x / the < TK 5 + 4x^2 now as X goes to positive Infinity 7 / X and 4 / x^2 each approach zero and so the limit of our function is X goes to positive Infinity is equal to 3 over the < TK 5 forx the absolute value of x * 1 /x is equal to -1 therefore our function forx = 3 - 7 /x / minus the < TK of 5 + 4x^2 now again 7 /x X and 4x^2 each go to zero as X goes to minus infinity and therefore as X goes to minus infinity F ofx approaches minus 3 over the < TK 5 one final example let's find the limit as X goes to Infinity of f ofx if f ofx is equal to the sare < TK of x + the < TK of x minus the < TK of x the first thing we'll do here is multiply our function by a fraction whose numerator and denominator are both equal to the same expression that defines F except with a plus sign between the two terms this lets us rewrite the function with a denominator of the < TK of X plus the < TK of X plus the < TK of X and with a numerator that's equal to x + the < TK of x - x so we simplify and obtain < TK of X over The < TK of x + the < TK X+ the < TK of X now let's multiply the numerator and denominator by 1 over the < TK of X this gives us 1 / the < TK of 1 + 1/ the < TK of x + 1 now since 1 over the < TK of X goes to zero as X goes to Infinity F ofx will approach 1 over 1 + 1 or 1 12 as X goes to Infinity