Limits involving infinity

I] Infinity, Minus Infinity as a limit as x approaches a: The line x=a is a vertical asymptote of the function: When lim f ( x), lim f ( x), or lim f ( x) is plus or minus infinity then the function f(x) has x  a x  a  x  a  the line x=a as a vertical asymptote.

Basic limits to remember: 1 1 1 lim   , lim   , x  0 is a vertical asymptote of f ( x)  x  0  x x  0  x x lim ln | x |  , x  0 is is a vertical asymptote of f ( x)  ln | x | x  0 x 2  3x  10 Example: Find each limit for the function f ( x )  . x 2  4 a) lim f ( x ) This is a rational function and substitution gives the indeterminate form 0/0. x  2 ( x  5)( x  2) x  5 7 We factor numerator and denominator.   as x  2 . ( x  2)( x  2) x  2 4 x  5 b) lim f ( x)  lim DNE because substitution gives 3/0. In fact the right side x  2 x  2 x  2 approaches positive infinity, the left side approaches minus infinity. The line x= -2 is a vertical asymptote.

When both sides go to positive infinity, we say the limit is infinity. Similarly if both sides go to negative infinity, the limit is minus infinity. 1 For example, we write lim   since both sides approach positive infinity. x  0 x 2 ln | x | Example: Find lim or show it does not exist. You can graph the function and see x  0 x its behavior near 0. ln | x | ln | x | ln | x | lim   lim   and lim does not exist.   x  0 x x  0 x x  0 x

II] Limits as x approaches positive infinity, and as x approaches minus infinity: Horizontal Asymptotes: When a function approaches one number, L, as x goes to infinity or to minus infinity, we say f(x) has horizontal asymptote y=L. There can be horizontal asymptotes if the function approaches one number as x goes to minus infinity and a different number as x goes to positive infinity. We will see functions that have horizontal asymptotes but first lets look at polynomials.

Polynomials never have horizontal asymptotes and always approach infinity or minus infinity as x goes to infinity or to minus infinity. The tendency of a polynomial as x approaches plus or minus infinity is the same as the tendency of its leading term. You can see this two ways: Graph Y 1  x 3  5 x 2  3x  10 Y 2  x 3 in your calculator. Use a window in which Xmin and Xmax are large, say Xmin=1000 and Xmax=1100. The graphs are nearly identical.

If x=1000, then the leading term is on the order of 1000 times any other term. lim x 3   so lim Y 1   The same works at minus infinity: x   x   lim x 3   so lim Y 1   x   x  

Ex. lim (2 x 5  20 x 4  15 x 3  35 )  lim  2 x 5   . Any odd power of x will approach x   x   minus infinity as x approaches minus infinity because an odd power of a negative is negative. Then the coefficient -2 makes it positive.

Rational functions: You only need to consider the tendency of the ratio of the leading terms. This only works for ratios. 2 x 4  15 x 3  16 x 2 2 x 4 2 1 Example: lim  lim   y=1/3 is a horizontal asymptote. x   6 x 4  32 x 3  50 x x   6 x 4 6 3

3x 2  4 x  10 3x 2 3 Example: lim  lim  lim  0 y=0 is a horizontal x   2 x 3  5 x 2  10 x  15 x   2 x 3 x   2 x asymptote.

4 x 4  3x 2  9 4 x 4 4 x Example: lim  lim  lim   There is no horizontal x   5 x 3  2 x 2 x   5 x 3 x   5 asymptote.

Other limits: "infinity minus infinity" is indeterminate.  2 2  Example: Find lim  x  4 x  x  1  Both terms tend to infinity so we have x     "infinity minus infinity".

( x 2  4 x  x 2  1 )( x 2  4 x  x 2  1 ) ( x 2  4 x )  ( x 2  1) x 2  4 x  x 2  1    2 2 2 2 x  4 x  x  1 x  4 x  x  1 4 x  1 4 x   2 x 2  4 x  x 2  1 2 x

Exercises: Show lim ( x 2  6 x  x )  3 and lim ( x 2  6 x  x )  3 x   x  

Show that every odd degree polynomial has at least one root. Use the Intermediate Value Theorem.