Y = (x + 1) * ln (x + 1) / x, when x infinitely approaches 0, what is the limit of Y? The answer is 1. I want to deduce the process

Y = (x + 1) * ln (x + 1) / x, when x infinitely approaches 0, what is the limit of Y? The answer is 1. I want to deduce the process

Click to enlarge, then click to enlarge

Limlnx / X when x tends to 0 + the limit is - ∞, limsin (x-1) / (x ^ 2-x) when x tends to 0 the limit is ∞, how to understand
Limlnx / X when x tends to 0 + the limit is - ∞, is it when x tends to 0, 1 / X tends to ∞, LNX tends to - ∞, they multiply to - ∞?
The limit of limsin (x-1) / (x ^ 2-x) when x tends to 0 is ∞. When x tends to 0, 1 / x, 1 / (x-1) tends to ∞, sin (x-1) is a constant. Are they multiplied by ∞?

Does limsin | x | / X exist when x tends to zero?

When x → 0 -
limsin|x|/x
=-limsinx/x
=-1
When x → 0 +
limsin|x|/x
=limsinx/x
=1
So there is no limit