INDETERMINATE FORMS AND L'HOSPITAL'S RULE

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,

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INDETERMINATE FORMS AND L'HOSPITAL'S RULE

The L'Hospital's Rule

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