Concept: Indeterminate Forms: Any expression whose value cannot be defined, like
‌,±‌,00,∞0 etc.
- L'Hospital's Rule: For the differentiable functions
f(x) and
g(x), the
‌, if
f(x) and
g(x)are both 0 or
±∞ (i.e. an Indeterminate Form) is equal to the
‌ if it exists.
- For the indeterminate form
∞−∞, first rationalize by multiplying with the conjugate and then divide the terms by the highest powerof the variable to get terms so that
‌→0 as
x→∞ - For the indeterminate form
‌, first try to rationalize by multiplying with the conjugate, or simplify by cancelling some terms in the numerator and denominator. Else, usethe L'Hospital's rule.
Calculation: ‌ ‌‌=‌ =‌, an indeterminate form.
Applying L'Hospital's rule, we get:
‌ ‌‌=‌| ax‌log‌a−ax2−1 |
| xx(log‌x+1) |
‌‌=‌| aa‌log‌a−a⋅aa−1 |
| an(log‌a+1) |
‌‌=‌ According to the question,
‌=−1.
⇒log‌a−1=−log‌a−1 ⇒2‌log‌a=0 ⇒a=1