Please help me with this L'Hospital's Rule

Question

Please help me with this question I am kinda confused on it

Answer 1

CORRECTION: In the last step before plugging in x=1 there should not be a 2 in the denominator but regardless the limit is still 5/2

Answer 2

The values for "x" are ≤ 1.

Ln(x) means to what power is used for 2.7182818..... Ln(10)=2.30258..... or e^2.30258....=10.

Base for natural logarithm.

Answer 3

Combine the terms to get lim_x→1 [(5xlnx - 5x + 5) / ((x-1)lnx)] which has the indeterminate form 0/0.

Applying L'Hopital's Rule, we get lim_x→1[(5lnx + 5 - 5) / (lnx + (x-1)/x)]= lim_x→1[5lnx / (lnx + 1 - 1/x)]

This limit also has the form 0/0. Apply LHopital again to get lim_x→1 [(5/x) / (1/x + 1/x²) =

lim_x→1 [(5/x) / ((x+1)/x²)] = lim_x→1[5x / (x+1)] = 5/2