Algebraic properties and limits

Calculation rules

For any real numbers a and b:

• e^(a+b) = e^a * e^b
• e^(-a) = 1 / e^a
• e^(a-b) = e^a / e^b
• (e^a)^n = e^(a*n), where n is an integer

These rules allow you to simplify expressions without a calculator.

Example

Let’s simplify A = e^(2x+3) / e^(x+3):

A = e^(2x+3-(x+3)) = e^(2x+3-x-3) = e^x

Derivative of a composite function

If u is a differentiable function, then (e^u)' = u' * e^u. For example, if f(x) = e^(3x+1), then f'(x) = 3 * e^(3x+1).

Reference limits

Limit	Result
lim x->+∞ e^x	+∞
lim x->−∞ e^x	0
lim x->+∞ e^x / x	+∞ (comparative growth)
lim x→−∞ x * e^x	0

These comparative growth limits mean that the exponential function always outweighs the powers of x as x approaches +∞.

Common pitfall

A common mistake is to write e^(a+b) = e^a + e^b: this is WRONG. The exponential function transforms sums into products, not the other way round. Similarly, e^(a*b) is not equal to e^a * e^b in general.

Key points

• e^(a+b) = e^a * e^b (never e^a + e^b).
• (e^u)' = u' * e^u.
• At +∞, exp(x) always dominates x^n.