L Hopital s rule

L Hopital s rule
Suppose that we have one of the following cases,
lim?(x?c)??f(x)/g(x) ?=( 0 )/( 0 ) or lim?(x?c)??f(x)/g(x) ?=( ?? )/( ?? )
where c can be any real number, infinity or negative infinity. In these cases we have,
lim?(x?c)??f(x)/g(x) ?=lim?(x?c)??(f^ (x))/(g^ (x) )?
Example: Evaluate the limits
1. lim?(x?0)??(e^x-x-1)/x^2 ? =0/( 0 )
lim?(x?0)??(e^x-x-1)/x^2 ?=lim?(x?0)??(e^x-1)/2x?=0/( 0 )
=lim?(x?0)??( e^x)/2?=( 1 )/( 2 )
2. lim?(x?0)??(2 sin?x-sin?2x)/(x-sin?x )? =0/( 0 )
lim?(x?0)??(2 sin?x-sin?2x)/(x-sin?x )?=lim?(x?0)??(2 cos?x-2 cos?2x)/(1-cos?x )?=0/( 0 )
=lim?(x?0)??(-2 sin?x+4 sin?2x)/sin?x ?=0/( 0 )
=lim?(x?0)??(-2 cos?x+8 cos?2x)/cos?x ?=(-2+8)/1=6
3. lim?(x?1/2)??((1-4x^2 ) sin^(-1)?x)/cos?(?x) ?=lim?(x?1/2)?sin^(-1)?x ×lim?(x?1/2)??(1-4x^2)/cos?(?x) ?=sin^(-1)?(1/2) lim?(x?1/2)??(-8x)/(-? sin?(?x) )?
=?/( 6 )×4/(? sin?(?/2) )=?/( 6 )×( 4 )/?=( 2 )/( 3 )
4. lim?(x??)??(5+ln?2x)/(x^2+4)?=?/?
lim?(x??)??(5+ln?2x)/(x^2+4)?=lim?(x??)??(( 2 )/2x)/2x=? lim?(x??)??1/(2x^2 )=? 1/?=0
5. lim?(x??)??(ln?x )^2/e^2x ?=?/?
lim?(x??)??(ln?x )^2/e^2x ?=lim?(x??)??(ln?x )^2/(e^x )^2 ?=(lim?(x??)??ln?x/e^x ? )^2=(lim?(x??)??(( 1 )/x)/e^x ? )^2=(0/?)^2=0
6. lim?(x?0)??(x^4 e^3x)/(e^2x-1)^2 ? =0/( 0 )
lim?(x?0)??(x^4 e^3x)/(e^2x-1)^2 ?=lim?(x?0)??(x^2 )^2/(e^2x-1)^2 ?×lim?(x?0) e^3x=(lim?(x?0)??x^2/((e^2x-1) )? )^2×1
=(lim?(x?0)??2x/(2e^2x )? )^2 =(lim?(x?0)??2/(4e^2x )? )^2=(( 2 )/4)^2=( 1 )/4


Evaluate the limits
1. lim?(x?0) ( ln?(1+x)-x)/(cos?x-1) 2. lim?(x?0) ( cos?4x-cos?2x)/x^2
3. lim?(x??)??(7-ln?(2x+3))/ln?(5x-2) ? 4. ?lim ??(x? -1) (1+cos?x)/(x^2+2x+1)
5. ?lim???(x?4)?? sin^2?(?x)/(e^(x-4)-x+3)? 6. lim?(x?0)?? x^3/(x-sinx)?
7. lim?(x?0)??(e^x-1-x)/(x sin?x )? 8. lim?(x??/4)??sin?4x/x(cos?x-sin?x ) ?