Question

lim x➝0± 3x-5/x

Answer 1

L[x:0,(3x-5)/(x)]

L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
L[x:0,(3x-5)/(x)]=L[x:0]( (d)/(dx) (3x-5))/( (d)/(dx) x)

Find the derivative of the numerator.
(d)/(dx) 3x-5=3

Find the derivative of the denominator.
(d)/(dx) x=1

The limit of 3 as x approaches 0 is 3
L[x:0,3]=3

The L[x:0,((3x-5))/(x)] is 3.
L[x:0,(3x-5)/(x)]=3

Answer 2

\[\lim_{x \rightarrow 0^+} \frac{3x-5}{x}=3-\frac{5}{x}=-\infty\]
\[\lim_{x \rightarrow 0^-} \frac{3x-5}{x}=3-\frac{5}{x}=+\infty\]

Answer 3

L`Hospital rule only valids when there is \[\frac \infty \infty \]
or
\[\frac00\]

Answer 4

ok good to know :)