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L'Hospital Rule

L'Hospital Rule

There are many techniques to calculate the limit of a function. But do you know that there exists a derivative method also to evaluate the limiting value of a function. L'Hospital's rule is used in calculus to determine the limits of indeterminate forms of the type 00, using derivatives.

L’Hospital’s rule many times helps in solving complex limit problems in an easier way.

Table of contents:

  • L'Hospital’s Rule
  • Practice problems
  • FAQs

L'Hospital’s Rule

L'Hospital's rule states that for functions f(x)andg(x) which are differentiable on an open interval I except possibly at a point c contained in I such that

If lim\belowxaf(x)g(x) takes 00, or form,then


This rule can be applied multiple times while evaluating the limit. The key concept is to differentiate the numerator and the denominator separately until the indeterminate form gets eliminated.

Note: For indeterminate forms other than 00,, , L’Hospital rule can be applied only after changing it to either 00 or form.

Example: Evaluate <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>&#x2192;</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>sin</mi><mi>x</mi></mrow><mi>x</mi></mfrac></math>

Here first we will check the indeterminate form of the given expression by putting x=0

sin 00=00 which clearly means that we can apply the L'Hospital Rule.

Now differentiating the numerator and denominator independently, we get

lim\belowx0cos x1 (Now, the indeterminate form is removed)

Hence, now directly we can evaluate the limiting value in the next step i.e.

lim\belowx0cos 01=11=1

Related Concept Video: L'Hospital Rule | Limits | Class 11 & 12 MATHS | JEE 2021/22 | JEE 2023 l Keshav Sir | BYJU'S JEE

Practice problems:

Example: Evaluate lim\belowx2sin(ex-2-1)ln(x-1)



Applying L'Hospital Rule

lim\belowx2cos (ex-2-1)× ex-21(x-1)

lim\belowx2cos (ex-2-1)× ex-2×(x-1)

lim\belowx2cos (e2-2-1)× e2-2×(2-1)

lim\belowx2cos (0)× e0×(1)=1


Example: If f(9)=4andf'(9)=3thenlim\belowx9f(x)-2x-3.


lim\belowx9f(x)-2x-3(00 form)

Now applying L'Hospital Rule




lim\belowx9f(x)-2x-3= 92

Example: If f(a)=2, f'(a)=1, g(a)=-1; g'(a)=2,thenlim\belowxag(x)f(a)-g(a)f(x)x-a.


lim\belowxag(x)f(a)-g(a)f(x)x-a(00 form)

Applying L'Hospital Rule




Example: If the graph of the function y=f(x) has a unique tangent at the point (a,0) through which the graph passes,then evaluate


<math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi>lim</mi><mo>&#xA0;</mo></mrow><mrow><mi>x</mi><mo>&#x2192;</mo><mi>a</mi></mrow></munder><mfrac><mrow><mo>&#xA0;</mo><msub><mi>log</mi><mi>e</mi></msub><mfenced><mrow><mn>1</mn><mo>+</mo><mn>6</mn><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced></mrow><mrow><mn>3</mn><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac><mspace linebreak="newline"/><mspace linebreak="newline"/><mspace linebreak="newline"/></math>


The graph passes through point (a,0) f(a)=0

Also, f(x) is differentiable at x=a ( Since, it has unique tangent at x=a )

Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi>lim</mi><mo>&#xA0;</mo></mrow><mrow><mi>x</mi><mo>&#x2192;</mo><mi>a</mi></mrow></munder><mfrac><mrow><mo>&#xA0;</mo><msub><mi>log</mi><mi>e</mi></msub><mfenced><mrow><mn>1</mn><mo>+</mo><mn>6</mn><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced></mrow><mrow><mn>3</mn><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac><mspace linebreak="newline"/><mspace linebreak="newline"/><mspace linebreak="newline"/></math> is of 00 form.Therefore, L'Hospital Rule can be applied




lim\belowx3x2-4x+35x-7( form)

Applying L'Hospital Rule

lim\belowx6x-4515lim\belowx6x-4 =lim\belowx3x2-4x+35x-7=(Notdefined)



lim\belowx3x-45x2-7( form)

Applying L'Hospital Rule




Example: Evaluatelim\belownn2n3.


lim\belownn2n3( form)

Weknown2=n(n+1)(2n+6)6lim\belownn(n+1)(2n+6)6n3lim\belown(n+1)(2n+6)6n2( form)


Question.1 Why does L'Hopital's rule not always work?

Answer: The L’ Hospital's Rule occasionally fails by becoming trapped in an endless loop. For example we might encounter a situation in which after applying the L’Hospital rule multiple times, the limit returns to the original limit, indicating that we will never reach to a conclusion.

Question.2 Can we use the standard formulas of limits and L'Hospital Rule simultaneously?

Answer: Yes the standard limit formulas can be used along with the L’Hospital Rule

Question.3 Why is this rule named L' Hospital?

Answer: The rule derives its name from a French mathematician that it why it has been named so.


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