Find the limit of (x → + ∞) LIM ((x + 5) ^ 0.5-x ^ 0.5), Can you write down the specific process?

Find the limit of (x → + ∞) LIM ((x + 5) ^ 0.5-x ^ 0.5), Can you write down the specific process?

Lim [x → + ∞] [√ (x + 5) - √ x], rationalization, up and down multiplied by [√ (x + 5) + x] simplification
=lim[x→+∞] [√(x+5)-√x][√(x+5)+√x]/[√(x+5)+√x]
=lim[x→+∞] (x+5-x)/[√(x+5)+√x]
=Lim [x → + ∞] 5 / [√ (x + 5) + √ x], because the denominator tends to be positive infinity, the whole fraction tends to zero
=0

Find the limit value of LIM (x - > 0) 3 ^ X

This is the limit of a continuous function
So just substitute x = 0
Answer = 1

LIM (1 / X-1 / e ^ x-1) x tends to 0+

LIM (1 / X-1 / e ^ x-1) x tends to 0+
=lim(x->0+)(e^x-1-x)/x(e^x-1)
=lim(x->0+)(e^x-1-x)/x²
=lim(x->0+)(e^x-1)/2x
=lim(x->0+)(e^x)/2
=1/2