INDETERMINATE FORMS AND L'HOSPITAL'S RULE

Subscribe to our ▶️ YouTube channel 🔴 for the latest videos, updates, and tips.

While compuiting the limits

of certain functions R(x), we may come across the following situations like,

We say that they have the form of a number. But values cannot be assigned to them in a way that is consistent with the usual rules of addition and mutiplication of numbers. We call these expressions indeterminate forms. Although they are not numbers, these indeterminate forms play a useful role in the limiting behaviour of a function.

John (Johann) Bernoulli discovered a rule using derivatives to compute the limits of fractions whose numerators and denominators both approach zero or ∞. The rule is known today as L’Hôspital’s Rule, named after Guillaume de L’Hospital’s, a French nobleman who wrote the earliest introductory differential calculus text, where the rule first appeared in print.

The L'Hospital's Rule

Let f(x) and g(x) be differentiable functions and g'(x) ≠ 0.

Problem 1 :

Solution :

Using L'Hospital's Rule,

Problem 2 :

Solution :

Using L'Hospital's Rule,

Again using L'Hospital's Rule,

Problem 3 :

Solution :

Let y = x^{sin x}.

y = x^{sin x}

Take natural logarithm on both sides.

ln y = ln x^{sin x}

ln y = sin x ⋅ ln x

Take the given limit on both sides.

Using L'Hospital's Rule,

Again using L'Hospital's Rule,

Problem 4 :

Solution :

Using L'Hospital's Rule,

Again using L'Hospital's Rule,

Problem 5 :

Solution :

Using L'Hospital's Rule,

Problem 6 :

Solution :

Using L'Hospital's Rule,

Again using L'Hospital's Rule,

Problem 7 :

Solution :

Using L'Hospital's Rule,

Subscribe to our ▶️ YouTube channel 🔴 for the latest videos, updates, and tips.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

About Us | Contact Us | Privacy Policy

INDETERMINATE FORMS AND L'HOSPITAL'S RULE

The L'Hospital's Rule

Recent Articles

10 Hard SAT Math Questions (Part - 40)

10 Hard SAT Math Questions (Part - 41)

ASTC Formula in Trigonometry