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Limit of Algebraic function, Algebraic Function of type ‘infinity/infinity' is considered one of the most asked concept.
311 Questions around this concept.
If $\lim _{x \rightarrow 0} \frac{3+\alpha \sin x+\beta \cos x+\log _e(1-x)}{3 \tan ^2 x}=\frac{1}{3}$, then $2 \alpha-\beta$ is equal to:
Limit of Algebraic function
(b) Factorization Method
In this method, we factorize numerators and denominators. The common factors are canceled out and the rest of the output is the final answer.
Illustration 1:
(c) Rationalization Method
This method is used when either numerator or denominator or both have fractional powers (like ½, ⅓ etc). After rationalization, the terms are factorized which on cancellation gives the final answer.
Let’s go through an illustration to understand better
Illustration 2:
Rationalizing numerator and denominator we get,
Limit of Algebraic Function Using Standard Result
As we studied in Binomial Theorem, the binomial expansion for any index
where, |x| < 1
When x is infinitely small (approaching to zero), we can ignore higher powers of x and can write (1 + x)n = 1 + nx (approximately).
We will use this fact to evaluate the value of
Let x = a + h,
Illustration 1:
Solution:
Algebraic Function of type ‘infinity/infinity’
To find the limit of the type x ➝ ∞, write the given expression in the form of (D ≠ 0), (N is numerator, D is denominator). Then divide both N and D by highest power of x occurring in both N and D to get a meaningful form.
An important result:
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