Guess the value of the limit (if it exists)

Question

Guess the value of the limit (if it exists) by evaluating the function at the given numbers:

x = 5.1, 5.05, 5.01, 5.001, 5.0001, 

... 4.9, 4.95, 4.99, 4.999, 4.9999

Guess the value of the limit (correct to six decimal places). (If an answer does not exist, enter DNE.)

Answer 1

No guessing ! Hah !

Recall the Definition of a Limit of f(x) at x₀ is not just about the value a function “tends” to at x₀, regardless of the actual value the function is defined to have at x₀, but f(x) must be defined on an "open interval" containing x₀, even if undefined at x₀ itself.

We cannot speak of a Limit without speaking of a suitable Open Interval for f(x) around x₀, in both directions around x₀.

On an open interval like (-4, 6), the limit does exists as x/(x + 5) = 5/10 = 1/2… (I will dispense with the decimals)

On the interval (-10, 10), the same limit does not exist, as a matter of definition, due to f(-5) being undefined on the specified open interval (DNE)

On (5, 10), the following "one-sided limit” (“right-handed limit") does exist: lim x->5⁺ (x² - 5x) / (x² - 25), but the limit does not. Why not ? Definitions… ! Which part ? (see above)

On (-5, 5), the following "one-sided limit” (“left-handed limit") does exist: lim x->5⁻ (x² - 5x) / (x² - 25), but the limit does not. Again, why not ? [Hint: It is not about the -5… why not ? And why does the limit not exist again ?]

Answer 2

Does the limit exist? Why or why not?