Limits of Exponential And Logarithmic Functions

Limits of Exponential And Logarithmic Functions

1. Exponential function : The functions of the form \( a^x ,e^x \), where x is a variable and a, e are constants are called exponential functions. e.g., f(x)=2^x, f(x)=e^x

2.  Some standard expansions
(i)\( e^x = 1 + x + \frac{{x^2 }} {{2!}} + \frac{{x^3 }} {{3!}} + \frac{{x^4 }} {{4!}} + ..... \)
(ii)\( e^{ - x} = 1 - x + \frac{{x^2 }} {{2!}} - \frac{{x^3 }} {{3!}} + \frac{{x^4 }} {{4!}}...... \)
(iii)\( a^x = 1 + x\left( {\log _e^a } \right) + \frac{{x^2 }} {{2!}}\left( {\log _e^a } \right)^2 + ...... \)
(iv)\( \log \left( {1 + x} \right) = x - \frac{{x^2 }} {2} + \frac{{x^3 }} {3} - \frac{{x^4 }} {4} + \frac{{x^5 }} {5}......( - 1 < x \leqslant 1) \)
(v)\( \log \left( {1 - x} \right) = - x - \frac{{x^2 }} {2} - \frac{{x^3 }} {3} - \frac{{x^4 }} {4} - \frac{{x^5 }} {5} + .....( - 1 \leqslant x < 1) \)
(vi)\( \sin x = x - \frac{{x^3 }} {{3!}} + \frac{{x^5 }} {{5!}} - \frac{{x^7 }} {{7!}} - \,\,\,.... \)
(vii)\( \cos x = 1 - \frac{{x^2 }} {{2!}} + \frac{{x^4 }} {{4!}} - \frac{{x^6 }} {{6!}} + ..... \)
(viii)\( \tan x = x + \frac{{x^3 }} {3} + \frac{2} {{15}}x^5 + ..... \)

Theorem : Prove that \( \mathop {Lt}\limits_{x \to 0} \frac{{e^x - 1}} {x} = 1 \)
Proof: we know,\( e^x = 1 + x + \frac{{x^2 }} {{2!}} + \frac{{x^3 }} {{3!}} + ..... \) 
\( \Rightarrow e^x - 1 = x + \frac{{x^2 }} {{2!}} + \frac{{x^3 }} {{3!}} + .... \)
\( \Rightarrow \frac{{e^x - 1}} {x} = 1 + \frac{x} {{2!}} + \frac{{x^2 }} {{3!}} + .... \)
\( \Rightarrow \mathop {Lt}\limits_{x \to 0} \frac{{e^x - 1}} {x} = \mathop {Lt}\limits_{x \to 0} \left( {1 + \frac{x} {{2!}} + \frac{{x^2 }} {{3!}} + .....} \right) \)
\( = 1 + \frac{0} {{2!}} + \frac{{0^2 }} {{3!}} + ...... = 1 \)
\( \therefore \mathop {Lt}\limits_{x \to 0} \frac{{e^x - 1}} {x} = 1 \) 

4. Prove that \( \mathop {Lt}\limits_{x \to 0} \frac{{a^x - 1}} {x} = \log _e^a (a > 0) \)
Proof:
We know \( a^x = 1 + x\left( {\log _e^a } \right) + \frac{{x^2 }} {{2!}}\left( {\log _e^a } \right)^2 + ..... \)
\( \Rightarrow a^x - 1 = x\left( {\log _e^a } \right) + \frac{{x^2 }} {{2!}}\left( {\log _e^a } \right)^2 + .... \)
\( \Rightarrow \frac{{a^x - 1}} {x} = \log _e^a + \frac{x} {{2!}}\left( {\log _e^a } \right)^2 + ..... \)
\( \Rightarrow \mathop {Lt}\limits_{x \to 0} \frac{{a^x - 1}} {x} = \mathop {Lt}\limits_{x \to 0} \left( {\log _e^n + \frac{x} {{2!}}\left( {\log _e^a } \right)^2 + .....} \right) \)
\( = \log _e^a + \frac{0} {{2!}}\left( {\log _e^a } \right)^2 + .... \)
\( = \log _e^a \)
\( Hence,\mathop {Lt}\limits_{x \to 0} \frac{{a^x - 1}} {x} = \log _e^a where\text{ }a > 0 \)

5. Some important formulae
i)\( \mathop {Lt}\limits_{x \to 0} \frac{{e^{ax} - 1}} {x} = a \)
ii)\( \mathop {Lt}\limits_{x \to 0} \frac{{e^{ax} - 1}} {{e^{bx} - 1}} = \frac{a} {b} \)
iii)\( \mathop {Lt}\limits_{x \to 0} \frac{{a^{\lambda x} - 1}} {x} = \lambda \cdot \log _e^a ,a > 0 \)
iv)\( \mathop {Lt}\limits_{x \to 0} \frac{{a^x - b^x }} {x} = \log _e \left( {\frac{a} {b}} \right) \)
v)\( \mathop {Lt}\limits_{x \to 0} \frac{{a^x - 1}} {{b^x - 1}} = \log _b^a \)
vi)\( \mathop {Lt}\limits_{x \to 0} \left( {1 + x} \right)^{\frac{1} {x}} = e \)
vii)\( \mathop {Lt}\limits_{x \to 0} \left( {1 + ax} \right)^{\frac{1} {x}} = e^a \)
viii)\( \mathop {Lt}\limits_{x \to \infty } \left( {1 + \frac{1} {x}} \right)^x = e \)
ix)\( \mathop {Lt}\limits_{x \to \infty } \left( {1 + \frac{a} {x}} \right)^x = e^a \)