L'Hospital's rule

L'Hospital's rule
Written by: Robert Pinterson

Earned my Ph.D. in mathematics from University of North Carolina at Chapel Hill.
I am a lecturer with over 5 years of teaching experience and an active researcher in the field of quantum information. Passionate about everything connected with maths in particular and science in general.

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L'Hospital's Rule

It was Introduced by French mathematician in 1600s.

L'hospital's rules apply only when we have 0/0 form or form.

Definition: It help us to find limit of functions when it is impossible to find.

Indeterminate forms

0x, ∞-∞, 0/0, ∞/∞, 00, 1, 0

Fact: Del Hospital Rule tells us when we have these above forms then we differentiate the function in numerator and denominator until we are out of these forms then apply limits.

General Form:

$$\lim\limits_{x \to k}\frac{f(x)}{g(x)}=\lim\limits_{x \to k}\frac{f'(x)}{g'(x)}$$

Fact: Here we take derivative of two functions by using dash(‘)

The limit as x approaches k of "f(x) over g(x)" equals the the limit as x approaches k of "f’(x) over g’(x)"

Example#1:

$$\lim\limits_{x \to 2}x^2+\frac{x−6}{x^2−4}$$

Solution:

$$\text{When}\;x=2$$

$$\lim\limits_{x \to 2}x^2+\frac{x−6}{x^2−4}=\frac{0}{0}\text{Form}$$

So,now we apply del hospital rule by differentiating top and down of function until we get out of this form:

$$\lim\limits_{x \to 2}\frac{(2x+1)}{2x}=\frac{5}{4}$$

Example#2:

$$\lim\limits_{x \to ∞}\frac{e^x}{x}$$

Solution:

$$\lim\limits_{x \to ∞}\frac{e^x}{x}$$

$$=( \frac{∞}{∞})\text{form}$$

Which is also indeterminate form so here apply del hospital rule

$$\lim\limits_{x \to ∞}\frac{e^x}{1}=∞$$

Example#3:

$$\lim\limits_{x \to ∞}\frac{lnx}{x}$$

Solution:

As (0/0) so differentiating numerator and denumerator

$$\lim\limits_{x \to ∞}\frac{\frac{1}{x}}{1}$$

Fact: Here we apply the limit as we are out of indeterminate form =0