Imagine you are driving a car towards a destination. The closer you get to your destination, the clearer you can see its details. In mathematics, limits of algebraic functions work in a similar way. Limits show the value approached by an algebraic function when its input variable approaches a certain value.
Algebraic functions are functions formed from combinations of algebraic operations such as addition, subtraction, multiplication, division, and exponentiation with rational exponents. Examples include polynomial functions like f(x)=x2+3x−2 and rational functions like .
For limits involving radical forms, we often need to rationalize. Direct substitution yields an indeterminate form.
Example: Calculate limx→1x−1x−1
Direct substitution:
1−11−1=01−1=00 (indeterminate form)
We rationalize by multiplying with the conjugatex+1. The purpose is to eliminate the radical form in the numerator using the formula (a−b)(a+b)=a2−b2:
Let's factor both. For x2+2x−15, we find two numbers that when multiplied give −15 and when added give 2. Those numbers are 5 and −3.
x2−9=(x−3)(x+3)
x2+2x−15=x2+5x−3x−15=x(x+5)−3(x+5)=(x−3)(x+5)
Therefore:
x→3lim(x−3)(x+5)(x−3)(x+3)=x→3limx+5x+3
Substitute x=3: 3+53+3=86=43
If the numerator is non-zero and the denominator is zero (like 0a with a=0), the limit approaches infinity. If both numerator and denominator are zero (the form 00), use factoring or rationalization techniques.
