Limits at Infinity and End Behavior Quiz

End Behavior

• Describes the behavior of a function as x approaches positive or negative infinity
• As x approaches positive infinity refers to the function's behavior as x becomes increasingly large
• As x approaches negative infinity refers to the function's behavior as x becomes increasingly small (more negative)
• End behavior is the term used to describe the behavior of a function as x increases or decreases without bound
• Horizontal asymptote is the line that the function approaches as x increases or decreases without bound
• The equation of the horizontal asymptote is y = L if a function approaches a specific number L as x increases or decreases without bound

Properties of Exponents

• b^0 = 1 for any non-zero real number b
• b^(p/q) = (b^(1/q))^p for any positive rational number p/q expressed in lowest terms
• b^(1/q) = qth root of b for any positive integer q
• b^(p/q) = qth root of b^p for any positive integers p and q
• b^n = (b^(1/q))^p for any positive integers p and q
• b^(p/q) = b^(m/n) = b^r, where r is the simplified rational number form of (mp)/(nq)
• b^m * b^n = b^(m+n) for any integers m and n
• (b^m)^n = b^(m*n) for any integers m and n
• (a*b)^n = a^n * b^n for any non-zero real numbers a and b and any integer n
• (a/b)^n = a^n / b^n for any non-zero real numbers a and b and any integer n
• (b^n)^m = b^(n*m) for any non-zero real numbers a and b and any integer n and m
• The assumption made throughout this section regarding the value of b is that b is a positive real number
• Irrational powers of b are defined by using limits
• Irrational powers of a positive real number b are defined by considering the powers of b that are rational numbers
• The limit of b^r as r approaches the irrational number is defined as the value of b raised to that irrational power
• The definition of a power with an irrational exponent can be extended to negative values of b, that is, b is a negative number and r is a rational number
• The value of a power with a negative base and a rational exponent is determined by the sign of the exponent
• If the exponent is even, the result is always positive
• If the exponent is odd, the result is always negative
• The value of a power with a negative base and an irrational exponent is also defined using limits
• Limits are used to define powers with irrational exponents because they allow us to take the limit of a sequence of rational numbers that approach the irrational number
• The limit of b^r as r approaches the irrational number is defined as the value of b raised to that irrational power