You CAN Ace Calculus

 precalculus - piecewise functions basic limits finite limits

Calculus Topics 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

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

free ideas to save on bags & supplies

17calculus > limits > one-sided limits

One-sided limits require a good understanding of piecewise functions. If you are a little rusty or just need a quick reminder, you can find a complete discussion of piecewise functions on the piecewise functions precalculus page.

One-sided limits are finite limits where we evaluate the limit from each side of a point individually. The notation we use is

 $$\displaystyle{ \lim_{x \to a^-}{f(x)} }$$ evaluate the limit on the left side of $$a$$, i.e. values of $$x < a$$ $$\displaystyle{ \lim_{x \to a^+}{f(x)} }$$ evaluate the limit on the right side of $$a$$, i.e. values of $$x > a$$

The negative and positive sign that look like exponents on the finite value $$a$$ indicate the side that we are looking at.

One of the reasons we need to look at limits on both sides of some number is when we are determining continuity. As you know from the continuity page, one of the requirements for continuity is that the limit at a point must exist. In order for a limit to exist, the limit from the left must be equal to the limit from the right, i.e. $$\displaystyle{\lim_{x \to a^-}{f(x)} = \lim_{x \to a^+}{f(x)}}$$.

Notice that we are NOT saying that the function value must be equal to the limit or even that the function need be defined at $$x=a$$, only that the limit be equal on both sides of a.

Here is a great video to build your intuition of one-sided limits. He uses an absolute value function to discuss the idea of one-sided limits and limits that do not exist.

Dr Chris Tisdell - Limit of a function [9min-11secs]

video by Dr Chris Tisdell

Here is a good video showing a graph with several one-sided limits.

Krista King Math - How to find limits on CRAZY GRAPHS [7min-47secs]

video by Krista King Math

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.

One-Sided Limits - Practice Problems Conversion

[A01-484] - [A02-485] - [A03-486] - [A04-487] - [A05-489] - [A06-490] - [A07-491] - [A08-495] - [A09-496]

[B01-488] - [B02-492] - [B03-494]

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

GOT IT. THANKS!

Instructions - - Unless otherwise instructed, evaluate the following limits. Give your answers in exact form.

Basic Problems

$$\displaystyle{\lim_{x\to1}{\frac{\abs{x-1}}{x-1}}}$$

Problem Statement

$$\displaystyle{\lim_{x\to1}{\frac{\abs{x-1}}{x-1}}}$$

Solution

484 video

video by PatrickJMT

$$\displaystyle{\lim_{x\to0^-}{\left(\frac{1}{x}-\frac{1}{\abs{x}}\right)}}$$

Problem Statement

$$\displaystyle{\lim_{x\to0^-}{\left(\frac{1}{x}-\frac{1}{\abs{x}}\right)}}$$

Solution

485 video

video by PatrickJMT

$$\displaystyle{\lim_{x\to5^+}{\frac{6}{x-5}}}$$

Problem Statement

$$\displaystyle{\lim_{x\to5^+}{\frac{6}{x-5}}}$$

Solution

486 video

video by PatrickJMT

$$\displaystyle{\lim_{x\to0}{\frac{x-1}{x^2(x+2)}}}$$

Problem Statement

$$\displaystyle{\lim_{x\to0}{\frac{x-1}{x^2(x+2)}}}$$

Solution

487 video

video by PatrickJMT

$$\displaystyle{\lim_{x\to-4^-}{\abs{x+4}}}$$

Problem Statement

$$\displaystyle{\lim_{x\to-4^-}{\abs{x+4}}}$$

Solution

489 video

video by PatrickJMT

$$\displaystyle{\lim_{x\to5^+}{\sqrt{x^2-25}}}$$

Problem Statement

$$\displaystyle{\lim_{x\to5^+}{\sqrt{x^2-25}}}$$

Solution

490 video

video by PatrickJMT

$$\displaystyle{\lim_{x\to5^-}{\sqrt{x(5-x)}}}$$

Problem Statement

$$\displaystyle{\lim_{x\to5^-}{\sqrt{x(5-x)}}}$$

Solution

491 video

video by Krista King Math

$$\displaystyle{\lim_{x\to0}{1/x}}$$

Problem Statement

$$\displaystyle{\lim_{x\to0}{1/x}}$$

Solution

495 video

$$\displaystyle{\lim_{x\to0}{1/x^2}}$$

Problem Statement

$$\displaystyle{\lim_{x\to0}{1/x^2}}$$

Solution

496 video

Intermediate Problems

Evaluate $$\displaystyle{\lim_{x\to1}{f(x)}}$$ for $$\displaystyle{f(x)=\left\{ \begin{array}{lr} x+3 & x \leq 1 \\ x^2-2x & x >1 \end{array}\right.}$$

Problem Statement

Evaluate $$\displaystyle{\lim_{x\to1}{f(x)}}$$ for $$\displaystyle{f(x)=\left\{ \begin{array}{lr} x+3 & x \leq 1 \\ x^2-2x & x >1 \end{array}\right.}$$

Solution

488 video

video by PatrickJMT

Prove that the limit $$\displaystyle{\lim_{x\to0}{\frac{\abs{x}}{x}}}$$ does not exist.

Problem Statement

Prove that the limit $$\displaystyle{\lim_{x\to0}{\frac{\abs{x}}{x}}}$$ does not exist.

Solution

492 video

video by Krista King Math

$$\displaystyle{\lim_{x\to0}{\frac{x-2\abs{x}}{\abs{x}}}}$$

Problem Statement

$$\displaystyle{\lim_{x\to0}{\frac{x-2\abs{x}}{\abs{x}}}}$$

Solution