Method of Evaluating Algebraic Limits
Method of evaluating Algebraic Limits when variable tends to Infinity: To evaluate this type of limits e follow the following procedure:
Step I: Write down the given expression in form of a rational function, i.e, \(\frac{f\left( x \right)}{g\left( x \right)}\), if it is not so.
Step II: If K is highest power of x in numerator and denominator both, then divide each term in numerator and denominator by xk.
Step III: Use the result \(\underset{x\to \infty }{\mathop{\lim}}\,\frac{1}{{{x}^{n}}}=0\), where n > 0.
An important result: If m, n are positive integers and a₀ b₀ ≠ 0 are non-zero real numbers, then \(\underset{x\to \infty }{\mathop{\lim}}\,\frac{{{a}_{0}}{{x}^{m}}+{{a}_{1}}{{x}^{m-1}}+….+{{a}_{m-1}}x+{{a}_{m}}}{{{b}_{0}}{{x}^{n}}+{{b}_{1}}{{x}^{n-1}}+….+{{b}_{n-1}}x+{{b}_{n}}}\).
\(\left\{ \begin{align}& \frac{{{a}_{0}}}{{{b}_{0}}},\,if\,\,m=n \\ & 0,\,if\,m<n \\& \infty ,\,if\,m>n\,\,and\,\,as\,\,{{b}_{0}}>0 \\ & -\infty ,\,if\,m>n\,and\,as\,{{b}_{0}}<0 \\\end{align} \right.\)
Evaluation of limits by using DE’L’ hospital’s rule: If f(x) and g(x) be two functions of x such that
- \(\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=0\)
- f(x) and g(x) both are continuous at x = a,
- f(x) and g(x) both are differentiable at x = a
- fˈ(x) and gˈ(x) are continuous at the point x = a, then \(\underset{x\to a}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to a}{\mathop{\lim }}\,\frac{f’\left( x \right)}{g’\left( x \right)}\), provided that g(a) ≠ 0.
Remark: The above rule is also applicable if \(\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=\infty \) and \(\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=\infty \).
Example: Evaluate \(\underset{x\to 2}{\mathop{\lim }}\,\frac{x-2}{{{x}^{2}}-4}\)
Solution: \(\underset{x\to 2}{\mathop{\lim }}\,\frac{x-2}{{{x}^{2}}-4}\)
Applying the L hospital’s rule
\(\underset{x\to 2}{\mathop{\lim }}\,\frac{1-0}{2x-0}\)
= 1/2 (1)
= ½