• HISTORY

Using Howard Zinn’s A People’s History of the United States, Chapter Seven: “As Long as Grass…

Chapter seven discusses the relationship between the young American government and the Native American populations. The chapter discusses the oppression of Native populations by the white…

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• HISTORY

What are three basic principles of American democracy?

In our Constitution, a democracy or a democratic republic was created. There are several principles of the American democracy. One principle is that our government is based on the concept of the…

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• A THEORY OF JUSTICE

Explain John Rawls’ theory of justice.

To put it simply, philosopher John Rawls defined justice as “fairness.” By this, he meant that the basic structures of society should be ordered in such a way that they promote maximum both…

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• HISTORY

What country did Ferdinand Magellan claim?

Ferdinand Magellan was a Portuguese explorer and the first person to plan a route for circumnavigating the globe. Unfortunately, he died before he could actually complete circumnavigation, but his…

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• HISTORY

Why did Socialism develop?

Socialism developed during the industrial revolutions of the nineteenth century. It was a reaction to the success of capitalism and the division of society into two distinct groups: the people who…

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• HISTORY

Why did the Tea Act of 1773 anger many American colonists?

The Tea Act of 1773 angered American political leaders for several reasons. First, it was seen as a sort of stalking horse for future regulations on American trade, which, while not necessarily as…

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• HISTORY

What are objects that we use now that the Victorians also used?

There are hundreds of common household items that were used both in Victorian times and in modern times. Some of these items were invented during the Victorian era (1837-1901) and are still used…

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• THANK YOU, M’AM

I need help with an essay on Thank you, M’am by Langston Hughes. It’s a persuasive essay about my…

Thank You, M’am by Langston Hughes serves as a guide to young adults. In writing a persuasive essay about it, the writer must convince the reader that a particular point of view is the best and…

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• SCIENCE

What are the advantages and disadvantages of using an iron catalyst?

A catalyst is a chemical compound that acts to help promote or speed up the rate of a chemical reaction. The catalyst itself is not incorporated into the product, it simply allows the reactants to…

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• MACBETH

What are the elements that influenced Macbeth to transform from a brave and loyal defender of the…

There are several elements that created Macbeth’s transformation from a loyal thane to a murderous tyrant. Here is one: the power of potential. It is common to discuss Macbeth’s transformation in…

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• SCIENCE

name 2 similarities between magnets and the atomic model

One similarity between magnets and the atomic model is the forces of attraction involved. Magnetic domains that are aligned so that their electromagnetic orientations are in opposite directions are…

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• MACBETH

How did Lady Macbeth and Macbeth respond after the murder of King Duncan? Why did Shakespeare…

After Lady Macbeth comes out of Duncan’s chamber, she’s talking to herself (so the audience can hear her thoughts and understand her emotions) about how she laid the daggers ready for Macbeth and…

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• LITERATURE

What is the summary for Station Eleven by Emily St. John Mandel?

Well, let us start with the main theme of the whole book: “survival is insufficient.” Everything from Mandel’s tale stems from there. In Station Eleven, we learn about the day that our…

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• MATH

limx→∞(2x−32x+5)2x+1limx→∞(2x-32x+5)2x+1 Find the limit. Use l’Hospital’s Rule…

Replacing ∞∞ for x in limit equation yields the nedetermination ∞∞∞∞ . You need to use the following technique, such that: f(x)=(2x−32x+5)2x+1f(x)=(2x-32x+5)2x+1

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• MATH

limx→0+(cos(x))1x2limx→0+(cos(x))1×2 Find the limit. Use l’Hospital’s Rule where…

You need to evaluate the limit, hence, you need to replace 0+0+ for x: limx→0+(cosx)1×2=(cos0)10+=1+∞limx→0+(cosx)1×2=(cos0)10+=1+∞ You may use the special limit

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• MATH

limx→1(2−x)tan(πx2)limx→1(2-x)tan(πx2) Find the limit. Use l’Hospital’s Rule where…

You need to evaluate the limit, hence, you need to replace 1 for x: limx→1(2−x)tan(π⋅x2)=(2−1)∞=1∞limx→1(2-x)tan(π⋅x2)=(2-1)∞=1∞ You need to use the logarithm special technique but first you need to…

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• MATH

limx→0+(4x+1)cot(x)limx→0+(4x+1)cot(x) Find the limit. Use l’Hospital’s Rule where…

Given the function limx→0+(4x+1)cot(x)limx→0+(4x+1)cot(x) We have to find the limit. The given function is an intermediate form of type 1∞1∞ which can be written as: i.e…

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• MATH

limx→∞(ex+x)1xlimx→∞(ex+x)1x Find the limit. Use l’Hospital’s Rule where…

Replacing ∞∞ for x in limit equation yields the nedetermination ∞o∞o . You need to use the following technique, such that: f(x)=(ex+x)1xf(x)=(ex+x)1xYou need to take logarithms both sides, such…

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• MATH

limx→∞x1xlimx→∞x1x Find the limit. Use l’Hospital’s Rule where appropriate. If…

Replacing ∞∞ for x in limit equation yields the nedetermination ∞o∞o . You need to use the logarithm technique, such that: f(x)=(x)1xf(x)=(x)1x You need to take logarithms both sides, such that:…

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• MATH

limx→∞xln(2)1+ln(x)limx→∞xln(2)1+ln(x) Find the limit. Use l’Hospital’s Rule where…

Given the limit function limx→∞x(ln(2)1+ln(x)limx→∞x(ln(2)1+ln(x) . We have to find the limit value. This is of the form ∞0∞0 and can be written as:…

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• MATH

limx→1+x11−xlimx→1+x11-x Find the limit. Use l’Hospital’s Rule where appropriate….

You need to evaluate the limit, using logarithm special technique, such that: f(x)=x11−xlnf(x)=ln(x11−x)f(x)=x11-xlnf(x)=ln(x11-x) lnf(x)=11−x⋅lnxlnf(x)=11-x⋅lnx limx→1+11−x⋅lnx=00limx→1+11-x⋅lnx=00 Since the…

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• MATH

limx→∞(1+ax)bxlimx→∞(1+ax)bx Find the limit. Use l’Hospital’s Rule where appropriate….

We will use the fact that limx→∞(1+1x)x=e.limx→∞(1+1x)x=e. limx→∞(1+ax)bx=limx→∞(1+ax)bxa⋅alimx→∞(1+ax)bx=limx→∞(1+ax)bxa⋅a Now we use substitution y=xa.y=xa. =limy→∞(1+1y)aby=limy→∞(1+1y)aby…

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• MATH

limx→0(1−2x)1xlimx→0(1-2x)1x Find the limit. Use l’Hospital’s Rule where appropriate. If…

You need to evaluate the given limit, hence, you need to replace 0 for x, such that: limx→0(1−2x)1x=(1−0)10=1∞limx→0(1-2x)1x=(1-0)10=1∞ You may use special limit limx→0(1+x)1x=elimx→0(1+x)1x=e…

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• MATH

limx→0+(tan(2x))limx→0+(tan(2x)) Find the limit. Use l’Hospital’s Rule where appropriate. If…

When x→0+,x→0+, 2x also →0+.→0+. And tan(y) is continuous at 0 as an elementary function inside its domain. Therefore limx→0+(tan(2x))=tan(0)limx→0+(tan(2x))=tan(0) = 0. l’Hospital’s Rule isn’t…

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• MATH

limx→0+xx√limx→0+xx Find the limit. Use l’Hospital’s Rule where appropriate. If…

Let’s transform the expression to get a form valid for use of l’Hospital’s Rule. xx√=(elnx)x√=ex√⋅lnx=e(lnxx−12)xx=(elnx)x=ex⋅lnx=e(lnxx-12) Consider the power,…

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• MATH

limx→1+[ln(x7−1)−ln(x5−1)]limx→1+[ln(x7-1)-ln(x5-1)] Find the limit. Use l’Hospital’s Rule…

You need to evaluate the limit but first you need to use the logaritmic property to convert the difference of logarithms into the logarithm of quotient:

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• MATH

limx→∞(x−ln(x))limx→∞(x-ln(x)) Find the limit. Use l’Hospital’s Rule where appropriate. If…

limx→∞x−ln(x)limx→∞x-ln(x) =limx→∞x(1−ln(x)x)=limx→∞x(1-ln(x)x) =limx→∞x(1−limx→∞ln(x)x)=limx→∞x(1-limx→∞ln(x)x) Now let us evaluate limx→∞ln(x)xlimx→∞ln(x)xApply L’Hospital rule , Test condition:…

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• MATH

limx→0(cot(x)−1x)limx→0(cot(x)-1x) Find the limit. Use l’Hospital’s Rule where appropriate….

limx→0cot(x)−1xlimx→0cot(x)-1x =limx→0cos(x)sin(x)−1x=limx→0cos(x)sin(x)-1x =limx→0xcos(x)−sin(x)xsin(x)=limx→0xcos(x)-sin(x)xsin(x) Apply L’Hospital rule , Test condition:0/0 =limx→0(xcos(x)−sin(x))'(xsin(x))’=limx→0(xcos(x)-sin(x))′(xsin(x))′…

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• MATH

limx→0+(1x−1ex−1)limx→0+(1x-1ex-1) Find the limit. Use l’Hospital’s Rule where…

limx→0+(1x−1ex−1)limx→0+(1x-1ex-1) Plug-in x=0 to the function. limx→0+(1x−1ex−1)=10−1e0−1=∞−∞limx→0+(1x-1ex-1)=10-1e0-1=∞-∞ Since the result is indeterminate, to take its limit apply L’Hospital’s…

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• MATH

limx→0(csc(x)−cot(x))limx→0(csc(x)-cot(x)) Find the limit. Use l’Hospital’s Rule where…

You need to evaluate the limit, such that: limx→0(cscx−cotx)=limx→01−cosxsinx=1−cos0sin0=1−10=00limx→0(cscx-cotx)=limx→01-cosxsinx=1-cos0sin0=1-10=00Since the limit is indeterminate 0000 , you may…

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• MATH

limx→1(xx−1−1ln(x))limx→1(xx-1-1ln(x)) Find the limit. Use l’Hospital’s Rule where…

limx→1(xx−1−1ln(x))limx→1(xx-1-1ln(x)) To solve, plug-in x=1. limx→1(xx−1−1ln(x))=11−1−1ln(1)=10−10=∞−∞limx→1(xx-1-1ln(x))=11-1-1ln(1)=10-10=∞-∞Since the result is indeterminate, we can apply the L’Hospital’s…

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• MATH

limx→(π2)−cos(x)sec(5x)limx→(π2)-cos(x)sec(5x) Find the limit. Use l’Hospital’s Rule where…

Here we can use more elementary method. I suppose that we know limx→0(sinxx)=1.limx→0(sinxx)=1. Let’s substitute x=π2−y,x=π2-y, then y→0y→0 .

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• MATH

limx→1+lnxtan(πx2)limx→1+lnxtan(πx2) Find the limit. Use l’Hospital’s Rule where…

You need to evaluate the limit, such that: limx→1+lnx⋅tan(π⋅x2)=0⋅(−∞)limx→1+lnx⋅tan(π⋅x2)=0⋅(-∞) You need to prepare the limit for l’Hospital’s rule, by writing it as a quotient:

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• MATH

limx→∞xtan(1x)limx→∞xtan(1x) Find the limit. Use l’Hospital’s Rule where appropriate. If…

limx→∞xtan(1x)limx→∞xtan(1x) =limx→∞tan(1x)1x=limx→∞tan(1x)1x Apply L’Hospital’s Rule, Test condition:0/0 =limx→∞tan(1x)'(1x)’=limx→∞tan(1x)′(1x)′ =limx→∞sec2(1x)(−1x−2)−1x−2=limx→∞sec2(1x)(-1x-2)-1x-2…

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• MATH

limx→∞x3e−x2limx→∞x3e-x2 Find the limit. Use l’Hospital’s Rule where appropriate….

limx→∞x3e−x2limx→∞x3e-x2 =limx→∞x3ex2=limx→∞x3ex2 Apply L’Hospital rule, =limx→∞(x3)'(ex2)’=limx→∞(x3)′(ex2)′ =limx→∞3x22xex2=limx→∞3x22xex2 =limx→∞3x2ex2=limx→∞3x2ex2 Again…

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• MATH

limx→0+sin(x)ln(x)limx→0+sin(x)ln(x) Find the limit. Use l’Hospital’s Rule where appropriate….

You need to evaluate the limit such that: limx→0+sinx⋅lnx=0⋅(−∞)limx→0+sinx⋅lnx=0⋅(-∞) You need to use the special limit limx→0+sinxx=1limx→0+sinxx=1 , such that:

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• MATH

limx→0cot(2x)sin(6x)limx→0cot(2x)sin(6x) Find the limit. Use l’Hospital’s Rule where appropriate….

limx→0cot(2x)sin(6x)limx→0cot(2x)sin(6x) The function cot(2x)sin(6x) is undefined at x=0. So to take its limit, let’s apply the L’Hospital’s Rule. To do so, express it as a rational function.

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• MATH

limx→∞x−−√e−x2limx→∞xe-x2 Find the limit. Use l’Hospital’s Rule where…

As x→+∞,x→+∞, x−−√→+∞x→+∞ and e−x2→0.e-x2→0. So the initial expression is an indeterminate form ∞⋅0.∞⋅0. Let’s transform it to use l’Hospital’s Rule:

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• MATH

limx→∞xsin(πx)limx→∞xsin(πx) Find the limit. Use l’Hospital’s Rule where appropriate. If…

limx→∞xsin(πx)limx→∞xsin(πx) =limx→∞sin(πx)1x=limx→∞sin(πx)1x Apply L’Hospital rule , Test condition:0/0 =limx→∞sin(πx)'(1x)’=limx→∞sin(πx)′(1x)′ =limx→∞cos(πx)(−πx−2)−1x−2=limx→∞cos(πx)(-πx-2)-1x-2…

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• MATH

limx→a+cos(x)ln(x−a)x−sin(x)limx→a+cos(x)ln(x-a)x-sin(x) Find the limit. Use l’Hospital’s Rule…

You need to evaluate the limit, hence, you need to replace a+a+ for x in limit, such that: limx→a+cosx⋅ln(x−a)x−sinx=cosa⋅ln(a−a)a−sinalimx→a+cosx⋅ln(x-a)x-sinx=cosa⋅ln(a-a)a-sinaNotice that

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• MATH

limx→0cos(x)−1+(12)x2x4limx→0cos(x)-1+(12)x2x4 Find the limit. Use l’Hospital’s Rule…

Given limit function is: limx→0cos(x)−1+(12)x2x4limx→0cos(x)-1+(12)x2x4 We have to find the limits. Applying the limit we see that, limx→0cos(x)−1+(12)x2x4=00limx→0cos(x)-1+(12)x2x4=00 So now applying the…

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• MATH

limx→0ex−e−x−2xx−sin(x)limx→0ex-e-x-2xx-sin(x) Find the limit. Use l’Hospital’s Rule…

Given the limit function limx→0ex−e−x−2xx−sin(x)limx→0ex-e-x-2xx-sin(x) . We have to find the limits. Applying the limits we get, limx→0ex−e−x−2xx−sin(x)=00limx→0ex-e-x-2xx-sin(x)=00 So we have to apply the…

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• MATH

limx→1xa−ax+a−1(x−1)2limx→1xa-ax+a-1(x-1)2 Find the limit. Use l’Hospital’s Rule where…

Given the limit limx→1xa−ax+a−1(x−1)2limx→1xa-ax+a-1(x-1)2 . We have to find the limit value. Applying the limits we get, limx→1=xa−ax+a−1(x−1)2=1a−a+a−1(1−1)2=00limx→1=xa-ax+a-1(x-1)2=1a-a+a-1(1-1)2=00 Since 1a=11a=1So…

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• MATH

limx→0+xx−1ln(x)+x−1limx→0+xx-1ln(x)+x-1 Find the limit. Use l’Hospital’s Rule where…

The denominator has limit −∞+0−1=−∞.-∞+0-1=-∞. The numerator has limit 0 (I’ll prove this below). Therefore there is no indeterminacy, 0∞=0.0∞=0. (zero is the answer). Now prove that xx→1xx→1 when…

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• MATH

limx→11−x+ln(x)1+cos(x)limx→11-x+ln(x)1+cos(x) Find the limit. Use l’Hospital’s Rule where…

limx→11−x+ln(x)1+cos(x)limx→11-x+ln(x)1+cos(x) To compute for its limit, plug-in x=1. =1−1+ln(1)1+cos(1)=1-1+ln(1)1+cos(1) =01+cos(1)=01+cos(1) =0=0 The result is finite value. This means that the function is defined at…

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• MATH

limx→0xtan−1(4x)limx→0xtan-1(4x) Find the limit. Use l’Hospital’s Rule where appropriate….

Given the limit limx→0xtan−1(4x)limx→0xtan-1(4x) . We have to find the limit value. Applying the limit we get, limx→0xtan−1(4x)=00limx→0xtan-1(4x)=00 So we use L’Hospital’s rule to obtain,…

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• MATH

limx→0x+sin(x)x+cos(x)limx→0x+sin(x)x+cos(x) Find the limit. Use l’Hospital’s Rule where…

limx→0x+sin(x)x+cos(x)limx→0x+sin(x)x+cos(x) plug in the value, =0+sin(0)0+cos(0)=0+sin(0)0+cos(0) =0+00+1=0+00+1 =01=01=0

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• MATH

limx→0cos(mx)−cos(nx)x2limx→0cos(mx)-cos(nx)x2 Find the limit. Use l’Hospital’s Rule where…

Given the limit limx→0cos(mx)−cos(nx)x2limx→0cos(mx)-cos(nx)x2 . We have to find the limit value Applying the limits we get, limx→0cos(mx)−cos(nx)x2=00limx→0cos(mx)-cos(nx)x2=00Using L’Hospital’s rule and then applying…

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• MATH

limx→0x3x3x−1limx→0x3x3x-1 Find the limit. Use l’Hospital’s Rule where…

limx→0x3x3x−1limx→0x3x3x-1 Apply L’Hospital rule , Test condition:0/0 =limx→0(x3x)'(3x−1)’=limx→0(x3x)′(3x-1)′ =limx→0x3xln(3)+3x3xln(3)=limx→0x3xln(3)+3x3xln(3) =limx→03x(xln(3)+1)3xln(3)=limx→03x(xln(3)+1)3xln(3)…

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• MATH

limx→∞(ln(x))2xlimx→∞(ln(x))2x Find the limit. Use l’Hospital’s Rule where appropriate….

You need to evaluate the limit, such that: limx→∞ln2xx=∞∞limx→∞ln2xx=∞∞ Since the limit is indeterminate ∞∞∞∞ , you may use l’Hospital’s rule: