L ' Hopital Law

The L ' Hopital law is the weapon we seek to limit. The L ' Hopital law is almost omnipotent and, of course, limited to the limits of type 0/0 and ∞/∞, when our general approach to generation is given an ∞/∞ limit of 0/0 or more. However, the power of the L ' Hopital law is very powerful if it fits with our wit, because many of the limits that cannot be obtained by means of surrogacy can be converted into a form that can be solved.

The proof of the law of L ' hopital is actually very simple. It is in the form of, when F (a) =g (a) =0, F (x) and G (x) near a 0, we divide the fraction up and down by x-a, obviously the fraction does not change, x-a is a volume approaching 0, according to the definition of the derivative can be obtained in the middle of the equation.

The direct application of the L ' hopital law is actually very easy to use, such as the Limit Lim (x→0) "X/sinx", is a 0/0-type limit, the fraction of the same derivative at the same time, get it equals Lim (x→0) "1/cosx", then we can use the surrogate method to get the limit value of 1. This limit is useful in engineering, such as the pendulum swing problem, we generally say the pendulum swing in the angle is not too large when the vibration is a harmonic, its basis is this limit: in the angle of θ close to 0, you can use θ instead of sinθθ, when the ball is linear, if the angle is too large, θ deviation from sinθθ too much, At this point the pendulum is not linear, and its motion is not a harmonic vibration.

As to the limit of shape like 0 ∞, it can be turned into 0 divided by 1/∞ form, that is, 0/0. Another very common example is the limit of a 1^∞. Lim (x→∞) "1+1/x" ^x=e, this limit is obtained by seeking the limit of ln "1+1/x" ^x, ln "1+1/x" ^x can also be converted into XLN "1+1/x", this is a 0 ∞ limit, the reader can try to prove it himself.

Summary: For an inability to directly find the limit, it into 0/0 or ∞/∞ form and then use the L ' hopital law will bring you surprise.

L ' Hopital Law