Create a function that has the following properties. Make sure to prove your function has these properties by evaluating the specific limits, and by showing your work and justification. a. The lim $→& 𝑓(𝑥) = D.N. E. Make sure to evaluate both the left sided limit and right sided limit. b. For the same function, the lim $→0 𝑓(𝑥) exists, but 𝑓(2) is undefined.
+130514 I'm assuming that in (a) you're asking to prove that the limit of the function as it approaches positive/neg infinity DNE....if so, we have this possibility
f(x) = x^2 / (x - 2)
lim x^2 / (x - 2) divide each term by x
x → inf
lim x / (1 - 2 / x) = ( inf ) / (1 - 2/ inf) = ( inf ) / ( 1 - 0) = (inf) / 1 = inf = DNE
x → inf
By the same token.......as lim f(x)→ neg inf = -inf / 1 = -inf = DNE
(b) The limit as x → 0 from both sides = 0.....but....the function does not exist at x = 2 because the denominator = 0 at that point
Here's the graph.......https://www.desmos.com/calculator/zb8jv6lzn8
