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Finding Limits Algebraically Without Direct Substitution

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

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Area Under Curves

Areas between curves

Average Value

Average value of a function

Basic Integration Techniques

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Concept of Limits

Finding limits from graphs

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits

Continuity

Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

Discontinuities of Rational Functions (denominator=0)

Critical Points and Extrema

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Optimization

Curve Analysis

Curve sketching

Derivative Concept

Estimating Derivatives from a table

Slope and equation of tangent line

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Derivative Rules

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivatives of Special Functions

Derivative of inverse trigonometric functions

Derivative of trigonometric functions

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

Fundamental Theorem of Calculus

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Implicit Differentiation

Implicit differentiation

Limits at Infinity

Infinite limits - vertical asymptotes

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Motion and Rates

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Riemann Sums

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

Tangent Lines

Linear approximation

Use of Linear Approximations in Estimating Square Roots

Linearization of Radical Functions

Linearization of Polynomial Functions

Linearization of Rational Functions

Linearization of Exponential Functions

Linearization of Logarithmic Functions

Linear Approximation of Trigonometric Functions

Parent Assignments

Challenge Zone

Foundation Review

math

Evaluating a limit by factoring a difference of squares

Evaluating a limit by expanding (5+h)² and cancelling h

Evaluating limit of sin(5x)/2x using the sine limit identity

Limit of |x|/x as x approaches zero using piecewise definition

Rationalizing the denominator to evaluate a limit with a square root

When direct substitution into a limit produces an indeterminate form like zero over zero, the function can often still be rewritten so the limit exists. This lesson covers factoring, rationalizing, and combining fractions to evaluate such limits algebraically, plus how to handle one-sided cases.

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Overview

Finding Limits Algebraically When Direct Substitution Fails

What You'll Learn

Direct substitution fails when plugging in the target value gives an indeterminate form such as zero over zero.

Factoring and cancelling common factors is the most common way to remove the trouble spot and reveal the real limit.

Rationalizing the numerator or denominator works well whenever a square root is causing the indeterminate form.

Combining fractions over a common denominator can simplify expressions with fractions inside fractions before you substitute.

The simplified expression agrees with the original function everywhere except at the single point causing the problem, which is exactly what a limit cares about.

One-sided limits use the same algebraic tools, just restricted to values approaching from the left or the right.

What You'll Practice

Factoring difference of squares to simplify rational functions

Expanding and canceling terms in polynomial limit expressions

Multiplying by conjugates to rationalize denominators with radicals

Applying the sine theta over theta limit formula with variable substitution

Evaluating limits of absolute value functions from both directions

Why This Matters

This Unit Includes

5 Video lessons

Practice exercises

Learning resources

Skills

Limits

Factoring

Conjugates

Rationalization

Trigonometric Limits

Piecewise Functions

One-Sided Limits

Indeterminate Forms