Question

Question about projecting limits for an even and odd theoretical function?

If f is a continuous function such that its limit as x approaches positive infinity is 5. Discuss the following limit: limit of f as x approaches negative infinity; for each condition below. If the limit exists find it. If not possible explain.

a) The graph of f is symmetric to the y-axis.

b) The graph of f is symmetric to the origin.

Answer 1

Check below.

Explanation:

• `lim_(xrarr+oo)f(x)=5` , `(A)`

`lim_(xrarr-oo)f(x)=` ?

`color(blue)(a))`

`f` symmetric to y axis `->` `f` even , for `x` `in` `RR` , `-x` `in` `RR` : `f(-x)=f(x)`

Example of even function `f(x)=x^2+1` graph{x^2+1 [-10, 10, -5, 5]}

• We set `x=-y`
  `x->-oo`
  `y->+oo`

`lim_(xrarr-oo)f(x)=` `lim_(yrarr+oo)f(-y)` `=` `lim_(yrarr+oo)f(y)` `=^((A))` `5`

`color(red)b)`

`f` symmetric to the origin `->` `f` odd , for `x` `in` `RR` , `-x` `in` `RR` : `f(-x)=-f(x)`

Example of an odd function `f(x)=x^3`
graph{x^3 [-10, 10, -5, 5]}

• We set `x=-y`
  `x->-oo`
  `y->+oo`

`lim_(xrarr-oo)f(x)=` `lim_(yrarr+oo)f(-y)` `=` `-lim_(yrarr+oo)f(y)` `=^((A))` `-5`