Using Series Expansions to Find Limits : still to do

Understanding the Basics

Series expansions are widely used in mathematics to approximate functions or to find the values approaching a limit.
Fundamental series expansions include Taylor series, and Maclaurin series, which are special cases of the Taylor series.
Understanding and identifying series expansions is vital to finding limits.

Series Expansions

The Taylor series for a function is an infinite series which approximates the function around a certain point.
The Maclaurin series is a special case of the Taylor series where the point of expansion is zero.
The general formula for a Maclaurin series is f(x) = f(0) + f’(0)x + f’‘(0)x^2/2! + f’’‘(0)x^3/3! + …
Each term of the series expansion involves higher derivatives of the function, with the nth term proportional to the nth derivative at the point of expansion.

Finding Limits Using Series Expansions

By using series expansions, the calculation of limits becomes simpler.
If a limit takes an indeterminate form such as 0/0 or ∞/∞, it often helps to use a series expansion of the function to resolve the indeterminacy.
A common technique is to expand both the numerator and the denominator as a series, then divide term by term.

Example Calculation

To calculate the limit as x approaches 0 of sin(x)/x:
1. Write the Maclaurin series for sin(x): sin(x) = x - x^3/3! + x^5/5! - …
2. Notice that x is a common factor that can be divided out, giving: 1 - x^2/3! + x^4/5! - …
3. As x → 0, the high-power terms all shrink to zero, and the limit is 1.

Key Points

Series expansions like Taylor and Maclaurin series can give a manageable way to find limits, by reducing functions to polynomial-like forms that can be divided term by term.
They are especially useful for functions that have indeterminate forms at the limit.
However, correctly using series expansions requires a good understanding of differentiation and the nature of series.