Find limits using power and root laws

Learning Objectives Apply the Power Law to evaluate the limit of a function raised to a positive integer power. Apply the Root Law to evaluate the limit of a root of a function. Combine the Power and Root Laws with other limit properties to solve multi-step limit problems. Evaluate limits of polynomial and rational functions using direct substitution as an application of the limit laws. Identify when the Root Law for an even root is applicable by checking if the limit of the radicand is non-negative. Evaluate limits of functions with rational exponents by interpreting them as an application of the Power and Root Laws. How do scientists model the decay of a radioactive substance as it approaches a stable state? They use limits with exponential functions! ⚛️ This tutorial foc...

TermDefinitionExample LimitA limit is the value that a function 'approaches' as the input 'approaches' some value. It describes the behavior of the function near a point, not necessarily at the point itself.The limit of f(x) = x^2 as x approaches 3 is 9. We write this as lim_{x \to 3} x^2 = 9. Direct SubstitutionFor continuous functions like polynomials and rational functions (where the denominator is not zero), the limit at a point can be found by simply substituting the input value into the function.To find lim_{x \to 2} (x^3 + 4), we can directly substitute x=2 to get (2)^3 + 4 = 8 + 4 = 12. Polynomial FunctionA function consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponent...

The Power Law for Limits lim_{x \to c} [f(x)]^n = [lim_{x \to c} f(x)]^n The limit of a function raised to a positive integer power 'n' is the limit of the function, itself raised to that power. This allows you to find the limit of the base function first, then apply the exponent. The Root Law for Limits lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{lim_{x \to c} f(x)} The limit of the nth root of a function is the nth root of the limit of that function. A critical condition applies: if 'n' is an even integer (like a square root), then the limit of the inner function, lim_{x \to c} f(x), must be greater than or equal to 0.