You CAN Ace Calculus

 precalculus basics of limits

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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17calculus > limits > epsilon-delta definition

The epsilon-delta definition of the limit is the formal mathematical definition of how the limit of a function at a point is formed. It is sometimes called the precise or formal definition of the limit. Here is what it looks like.

$$\epsilon$$-$$\delta$$ Limit Definition

Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement $\lim_{x \rightarrow c}{~f(x)}=L$ means that for each $$\epsilon \gt 0$$ there exists a $$\delta \gt 0$$ such that if $$0 \lt \left|x-c\right| \lt \delta$$, then $$\left|f(x)-L\right| \lt \epsilon .$$

This is called the epsilon-delta definition of the limit because of the use of $$\epsilon$$ (epsilon) and $$\delta$$ (delta) in the text above. This is standard notation that most mathematicians use, so you need to use it as well. See the Use Of The Greek Alphabet in Mathematics section on the notation page for more information.

Something that may not be obvious from the definition is that $$\delta$$ depends on $$\epsilon$$, i.e. $$\delta = M(\epsilon)$$ where $$M$$ is some function. The third video below emphasizes this and shows details on how this works.

This definition is not easy to get your head around and it takes some thinking, working practice problems and time. I highly recommend you watch this video. It will help you get a feel for the concept of this definition.

### Krista King Math - Limit Definition [6min-41secs]

video by Krista King Math

Okay, after watching the above video, you should have at least an idea of what the epsilon-delta definition of the limit means. Now let's watch a video that discusses this in more depth.

### PatrickJMT - Precise Definition of a Limit - Understanding the Definition [11min-5secs]

video by PatrickJMT

This next video is a great video showing an example. Normally, we would make a practice problem from this video. However, this video has some excellent explanation that we don't want you to miss.

### Dr Chris Tisdell - Limit Definition [6min-22secs]

In this video, he talks about the $$\epsilon$$-$$M$$ definition and describes $$M$$ as a function of $$\epsilon$$. His $$M$$ is what we call $$\delta$$ and our experience is that most books and instructors use $$\delta$$. However, his use of $$M(\epsilon)$$ is good since it emphasizes that $$\delta$$ is a function of (and depends on) $$\epsilon$$.

video by Dr Chris Tisdell

Okay, time for a few practice problems. After that, you will be ready for the next topic, finite limits.

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Practice Problems Conversion

[A01-447] - [A02-2014] - [A03-2017] - [B01-448] - [B02-449] - [B03-450]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

GOT IT. THANKS!

Basic Problems

Use the $$\epsilon$$-$$\delta$$ definition of the limit to prove $$\displaystyle{\lim_{x\to-3}{(7x-9)}=-30}$$.

Problem Statement

Use the $$\epsilon$$-$$\delta$$ definition of the limit to prove $$\displaystyle{\lim_{x\to-3}{(7x-9)}=-30}$$.

Solution

### 447 solution video

video by Krista King Math

Use the $$\epsilon$$-$$\delta$$ definition of the limit to prove $$\displaystyle{\lim_{x\to 4}{(2x+3)} = 11}$$

Problem Statement

Use the $$\epsilon$$-$$\delta$$ definition of the limit to prove $$\displaystyle{\lim_{x\to 4}{(2x+3)} = 11}$$

Solution

### 2014 solution video

video by PatrickJMT

Use the $$\epsilon$$-$$\delta$$ definition of the limit to prove $$\displaystyle{\lim_{x\to 2}{(x^2-4x+5)}=1}$$

Problem Statement

Use the $$\epsilon$$-$$\delta$$ definition of the limit to prove $$\displaystyle{\lim_{x\to 2}{(x^2-4x+5)}=1}$$

Solution

### 2017 solution video

video by PatrickJMT

Intermediate Problems

Given the limit $$\displaystyle{\lim_{x\to1}{(2x+3)}=5}$$, find the largest value of $$\delta$$ such that $$\epsilon < 0.01$$.

Problem Statement

Given the limit $$\displaystyle{\lim_{x\to1}{(2x+3)}=5}$$, find the largest value of $$\delta$$ such that $$\epsilon < 0.01$$.

Solution

### 448 solution video

video by PatrickJMT

Given the limit $$\displaystyle{\lim_{x\to-1}{(3x+1)}=-2}$$, find the largest value of $$\delta$$ such that $$\epsilon=0.01$$.

Problem Statement

Given the limit $$\displaystyle{\lim_{x\to-1}{(3x+1)}=-2}$$, find the largest value of $$\delta$$ such that $$\epsilon=0.01$$.

Solution

### 449 solution video

video by PatrickJMT

Find $$\displaystyle{\lim_{x\to1}{\frac{3x(x-1)}{x-1}}}$$ and prove your answer using the $$\epsilon$$-$$\delta$$ definition.

Problem Statement

Find $$\displaystyle{\lim_{x\to1}{\frac{3x(x-1)}{x-1}}}$$ and prove your answer using the $$\epsilon$$-$$\delta$$ definition.

Solution