Evaluation and Algebric of Limits

The algebra of limits: Let f and g be two real functions with domain D. We define four new functions f ± g, fg, f/g on domain D by setting

(f ± g) (x) = f(x) ± g(x), (fg) (x) = f(x) g(x)

(f/ g)(x) = f(x)/ g(x), if g(x) ≠ 0 for any x ϵ D.

Let $$\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=l$$ and$$\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=m$$  if l and m exist, then

1. $$\underset{x\to a}{\mathop{\lim }}\,\left( f\pm g \right)\left( x \right)=\,\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)\pm \,\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=l\pm m$$

2. $$\underset{x\to a}{\mathop{\lim }}\,\left( fg \right)\left( x \right)=\underset{x\to a}{\mathop{\lim }}\,f\left( x \right).\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=lm$$

3. $$\underset{x\to a}{\mathop{\lim }}\,\left( \frac{f}{g} \right)\left( x \right)=\frac{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}{\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)}=\frac{l}{m}$$, provided

4. $$\underset{x\to a}{\mathop{\lim }}\,Kf\left( x \right)=K.\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$$, where K is constant

5. $$\underset{x\to a}{\mathop{\lim }}\,\left| f\left( x \right) \right|=\left| \underset{x\to a}{\mathop{\lim }}\,f\left( x \right) \right|=\left| l \right|$$

6. $$\underset{x\to a}{\mathop{\lim }}\,{{\left[ f\left( x \right) \right]}^{g\left( x \right)}}={{l}^{m}}$$

7. f (x) ≤ g (x) for every x in the deleted nbd of a, then $$\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)\le \underset{x\to a}{\mathop{\lim }}\,g\left( x \right)$$

8. f(x) ≤ g (x) ≤ h for every x in the deleted ndb of a and $$\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=l=\underset{x\to a}{\mathop{\lim }}\,h\left( x \right)$$ then $$\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=l$$.

9. $$\underset{x\to a}{\mathop{\lim }}\,fog\left( x \right)=f\left( \underset{x\to a}{\mathop{\lim }}\,f\left( x \right) \right)=\log l$$

In particular:

i) $$\underset{x\to a}{\mathop{\lim }}\,\log f\left( x \right)=\log \left( \underset{x\to a}{\mathop{\lim }}\,f\left( x \right) \right)=\log l$$

ii) $$\underset{x\to a}{\mathop{\lim }}\,{{e}^{f\left( x \right)}}={{e}^{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}}={{e}^{l}}$$

iii) If $$\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=$$ + or – ∞, then $$\underset{x\to a}{\mathop{\lim }}\,\frac{1}{f\left( x \right)}=0$$.

Evaluation of limits:  In the previous sections, we have discussed the notion of left hand limit (LHL), right hand limit (RHL) and the existence of the limit of a function f (x) at a given point. In the evaluation of limits, it is assumed that $$\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$$ always exists i.e., $$\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)$$.In this section, we will discuss various methods of evaluating limits. To facilitate the job of evaluation of limits we categorize problems on limits in the following categories:

(i) Algebraic limits

(ii) Trigonometric limits

(iii) Exponential and logarithmic limits.