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.
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,
Again using L'Hospital's Rule,
Problem 3 :
Solution :
Let y = xsin x.
y = xsin x
Take natural logarithm on both sides.
ln y = ln xsin 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,