You CAN Ace Calculus

### Topics You Need To Understand For This Page

 precalculus basics of 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

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17calculus > limits > pinching theorem

 Pinching Theorem Pinching Theorem At Infinity Practice

The Pinching Theorem is also referred to as The Sandwich Theorem or The Squeeze Theorem.

Pinching Theorem

For an interval I containing a point a, if we have three functions, $$f(x), g(x), h(x)$$ all three defined on the interval I, except at a and we have

$$\displaystyle{ g(x) \leq f(x) \leq h(x) }$$

for all points in I except possibly at $$x=a$$ and we know that

$$\displaystyle{ \lim_{x \to a}{g(x)} = \lim_{x \to a}{h(x)} = L }$$

Then $$\displaystyle{ \lim_{x \to a}{f(x)} = L }$$

The Pinching Theorem is a powerful theorem that allows us to determine several important limits, including this important trig limit.

 $$\displaystyle{ \lim_{\theta \to 0}{\frac{\sin(\theta)}{\theta}} = 1 }$$

### Proof of sin(x)/x Limit

prove that

$$\displaystyle{ \lim_{\theta \to 0}{\frac{\sin(\theta)}{\theta}} = 1 }$$

An informal proof is shown in this video.

### Khan Academy - Proof of sin(x)/x Limit [18min-4secs]

video by Khan Academy

To help you get your head around this theorem, here is a graph that intuitively shows you the idea of the proof of the above limit. The red line is the function $$\color{#BF003F}{ f(x) = \sin(x)/x }$$ , the green line is $$\color{#007F7F}{g(x) = \cos(x) }$$ and the blue line is $$\color{#0000FF}{h(x) = 1}$$. Notice that near, i.e. within an interval, around $$x=0$$,
$$\color{#007F7F}{g(x)} \leq \color{#BF003F}{f(x)} \leq \color{#0000FF}{h(x)}$$ and

 $$\displaystyle{ \color{#007F7F}{\lim_{x \to 0}{g(x)} = 1} }$$ $$\displaystyle{ \color{#0000FF}{\lim_{x \to 0}{h(x)} = 1} }$$

which means $$\displaystyle{ \color{#BF003F}{\lim_{x \to 0}{f(x)} = 1} }$$

The trick comes in when you have to find two functions $$g(x)$$ and $$h(x)$$ that satisfy the theorem. However, if you graph $$f(x)$$, sometimes you will be able to find a couple of functions that work. However, there is one idea that will sometimes work. If you have the trig functions $$\sin(x)$$ or $$\cos(x)$$, these functions are always between $$-1$$ and $$+1$$. So setting up an inequality with those will sometimes give you a place to start.

Okay, let's look at some videos to explain this in more detail. The first video explains the pinching(squeeze) theorem with a graph and it is a good place to start.

### PatrickJMT - The Squeeze Theorem for Limits, Example 1 [7min-12secs]

video by PatrickJMT

Here is a great video showing the proof of $$\displaystyle{ \lim_{x \to 0}{\left[ x\sin(1/x)\right]} = 0 }$$ using the pinching theorem. He does some unusual things here, so it is important to watch this video to see this technique.

### Dr Chris Tisdell - Limit of a function: Pinching theorem with streamlined method of solution [4min-22secs]

video by Dr Chris Tisdell

Pinching Theorem At Infinity

In the above discussion, we used the pinching theorem when the limit variable was going to a finite number. However, we can apply the same technique when the limit variable is going to infinity, i.e. we have the limit $$\displaystyle{ \lim_{x \to \infty}{f(x)} }$$. We just need to show that the function is bounded above and below by two functions and that the limits of both bounding functions goes to the same value as x goes to infinity. You will find some practice problems below demonstrating this technique.

### 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.

Pinching Theorem - Practice Problems Conversion

[A01-470] - [A02-472] - [A03-477] - [A04-1948]

[B01-471] - [B02-473] - [B03-474] - [B04-475] - [B05-476] - [B06-478] - [B07-479] - [B08-480] - [B09-481]

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

GOT IT. THANKS!

Instructions - Unless otherwise stated, evaluate the following limits using the pinching theorem. Give your answers in exact form.

Basic Problems

If $$3x \leq f(x) \leq x^3+2$$ on $$[0,2]$$, evaluate $$\displaystyle{ \lim_{x \to 1}{f(x)} }$$.

Problem Statement

If $$3x \leq f(x) \leq x^3+2$$ on $$[0,2]$$, evaluate $$\displaystyle{ \lim_{x \to 1}{f(x)} }$$.

Solution

### 470 video

video by PatrickJMT

If $$4\leq f(x)\leq x^2+6x-3$$ for all $$x$$, evaluate $$\displaystyle{ \lim_{x \to 1}{f(x)} }$$.

Problem Statement

If $$4\leq f(x)\leq x^2+6x-3$$ for all $$x$$, evaluate $$\displaystyle{ \lim_{x \to 1}{f(x)} }$$.

Solution

### 472 video

video by PatrickJMT

Evaluate $$\displaystyle{\lim_{x\to4}{f(x)}}$$ if, for all x, $$4x-9\leq f(x)\leq x^2-4x+7$$.

Problem Statement

Evaluate $$\displaystyle{\lim_{x\to4}{f(x)}}$$ if, for all x, $$4x-9\leq f(x)\leq x^2-4x+7$$.

Solution

### 477 video

video by Krista King Math

$$\displaystyle{ \lim_{n\to\infty}{\frac{\cos^2n}{n}}}$$

Problem Statement

$$\displaystyle{ \lim_{n\to\infty}{\frac{\cos^2n}{n}}}$$

Final Answer

$$\displaystyle{ \lim_{n\to\infty}{\frac{\cos^2n}{n}}=0}$$

Problem Statement

$$\displaystyle{ \lim_{n\to\infty}{\frac{\cos^2n}{n}}}$$

Solution

### 1948 video

video by Dr Chris Tisdell

Final Answer

$$\displaystyle{ \lim_{n\to\infty}{\frac{\cos^2n}{n}}=0}$$

Intermediate Problems

$$\displaystyle{\lim_{x\to0}{\left[x^2\cdot\cos(1/x^2)\right]}}$$

Problem Statement

$$\displaystyle{\lim_{x\to0}{\left[x^2\cdot\cos(1/x^2)\right]}}$$

Solution

### 471 video

video by PatrickJMT

$$\displaystyle{\lim_{x\to0}{x^4\sin(3/x)}}$$

Problem Statement

$$\displaystyle{\lim_{x\to0}{x^4\sin(3/x)}}$$

Solution

### 473 video

video by PatrickJMT

$$\displaystyle{\lim_{n\to\infty}{\frac{1}{n^3}\sin(n^2)}}$$

Problem Statement

$$\displaystyle{\lim_{n\to\infty}{\frac{1}{n^3}\sin(n^2)}}$$

Solution

### 474 video

video by PatrickJMT

$$\displaystyle{\lim_{n\to\infty}{\frac{(-1)^n+n^2}{n^2}}}$$

Problem Statement

$$\displaystyle{\lim_{n\to\infty}{\frac{(-1)^n+n^2}{n^2}}}$$

Solution

### 475 video

video by PatrickJMT

$$\displaystyle{\lim_{n\to\infty}{n^{-n^3}}}$$

Problem Statement

$$\displaystyle{\lim_{n\to\infty}{n^{-n^3}}}$$

Solution

### 476 video

video by PatrickJMT

$$\displaystyle{\lim_{x\to0}{x^2\cos(10x)}}$$

Problem Statement

$$\displaystyle{\lim_{x\to0}{x^2\cos(10x)}}$$

Solution

### 478 video

video by Krista King Math

$$\displaystyle{\lim_{x\to0}{x^2\cos(1/\sqrt[3]{x})}}$$

Problem Statement

$$\displaystyle{\lim_{x\to0}{x^2\cos(1/\sqrt[3]{x})}}$$

Solution

### 479 video

video by Krista King Math

$$\displaystyle{\lim_{x\to0}{\sqrt[3]{x}\sin(1/x)}}$$

Problem Statement

$$\displaystyle{\lim_{x\to0}{\sqrt[3]{x}\sin(1/x)}}$$

Solution

### 480 video

video by Krista King Math

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

Problem Statement

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

Solution

### 481 video

video by Krista King Math