Find the left and right limits of F (x) = x / x, φ (x) = | x | / X when x → 0, and explain whether their limits exist when x → 0

Find the left and right limits of F (x) = x / x, φ (x) = | x | / X when x → 0, and explain whether their limits exist when x → 0

The existence of left limit 1 and right limit 1 of F (x)
G (x) right limit 1 right limit-1 does not exist

Let f (x) be continuous at x = 0 and f (x) / X limit exist when x approaches 0. It is proved that f (x) is continuous and differentiable at x = 0
Why does limf (x) / X exist, denominator -- > 0, so limf (x) = 0?

Because if limf (x) is not equal to 0, the limit of F (x) / X does not exist
Let limf (x) = C ≠ 0
Then f (x) / X tends to + ∞ or - ∞ when X - > 0
That is, f (x) / X limit does not exist

It is proved that the necessary and sufficient condition for the existence of the limit of F (x) is that its left and right limits at X. exist and are equal

Necessity
① Given (x → x0) limf (x) = a, then
Let ε > 0
There is always δ > 0
When 0