Limits - Math

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Question

Evaluate the limit below:

$y=\lim_{x \to\frac{\pi }2{} }(sinx-cosx)^{tanx}$

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Answer

will approach $1^{\infty }$ when approaches $\frac{\pi }{2}$ , so $\lim_{x\rightarrow \frac{\pi }{2}} (lny)$ will be of type $0\cdot \infty$ as shown below:

$\lim_{x\rightarrow \frac{\pi }{2}} lny = \lim_{x\rightarrow \frac{\pi }{2}} ln (sinx-cosx)^{tanx}=\lim_{x\rightarrow \frac{\pi }{2}} tanx\cdot ln(sinx-cosx) = \infty \cdot 0$

So, we can apply the L’ Hospital's Rule:

$\lim_{x\rightarrow \frac{\pi }{2}} lny = \lim_{x\rightarrow \frac{\pi }{2}} \frac{sinx \cdot ln(sinx-cosx)}{cosx}=$

$\lim_{x\rightarrow \frac{\pi }{2}} \frac{cosx\cdot ln(sinx-cosx)+sinx\cdot (\frac{cosx+sinx}{sinx-cosx})}{-sinx}=\frac{0+1}{-1}=-1$

since:

$\lim_{x\rightarrow \frac{\pi }{2}} lny=-1$

hence:

$\lim_{x\rightarrow \frac{\pi }{2}} y=\lim_{x\rightarrow \frac{\pi }{2}} e^{lny}=e^{-1}=\frac{1}{e}$

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Question

The speed of a car traveling on the highway is given by the following function of time:

What can you say about the car's speed after a long time (that is, as approaches infinity)?

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Answer

The function given is a polynomial with a term , such that is greater than 1.

Whenever this is the case, we can say that the whole function diverges (approaches infinity) in the limit as approaches infinity.

This tells us that the given function is not a very realistic description of a car's speed for large !

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Question

$\textup{Find: }\lim_{x \to \infty }\frac{2x^{2}-7x}{x^{2}+3}$

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Answer

$\textup{As } x \to\infty,\textup{all but the terms with the highest exponents become insignificant.}$

$\textup{As }x^{2} \textup{ is the highest on top and bottom, evaluate the coefficients:}} \frac{2}{1}=2$

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Question

Calculate $\lim_{x\rightarrow \infty }\left ( x^{2} \sin \frac{1}{ x^{2}-1} - \sin \frac{1}{ x^{2}-1} \right )$ .

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Answer

This can be rewritten as follows:

$\lim_{x\rightarrow \infty }\left ( x^{2} \sin \frac{1}{ x^{2}-1} - \sin \frac{1}{ x^{2}-1} \right )$

$= \lim_{x\rightarrow \infty }\left ( x^{2}\cdot \sin \frac{1}{ x^{2}-1} -1\cdot \sin \frac{1}{ x^{2}-1} \right )$

$= \lim_{x\rightarrow \infty }\left [\left ( x^{2}-1 \right ) \cdot \sin \frac{1}{ x^{2}-1} \right]$

$= \lim_{x\rightarrow \infty } \frac{ \sin \frac{1}{ x^{2}-1} }{\frac{1}{ x^{2}-1}}$

We can substitute $u = \frac{1}{ x^{2}-1}$ , noting that as $x\rightarrow \infty$ , $u \rightarrow 0$ :

$= \lim_{u\rightarrow 0} \frac{ \sin u }{u} = 1$ , which is the correct choice.

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Question

Let $f(x)=\frac{1}{x}$ .

Find $\lim_{x\ \to \0^{-}} \frac{1}{x}$ .

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Answer

This is a graph of $\frac{1}{x}$ . We know that $\frac{1}{0}$ is undefined; therefore, there is no value for . But as we take a look at the graph, we can see that as approaches 0 from the left, approaches negative infinity.

This can be illustrated by thinking of small negative numbers.

NOTE: Pay attention to one-sided limit specifications, as it is easy to pick the wrong answer choice if you're not careful.

$\lim_{x \to \0^+} \frac{1}{x}$ is actually infinity, not negative infinity.

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Question

$f(x)=\frac{x^{2}-x-6}{x+2}$

$\lim_{x\rightarrow -2}f(x) =$

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Answer

A limit describes what -value a function approaches as approaches a certain value (in this case, ). The easiest way to find what -value a function approaches is to substitute the -value into the equation.

$f(-2)=\frac{(-2)^{2}-(-2)-6}{-2+2}=\frac{4+2-6}{0}=\frac{0}{0}$

Substituting for gives us an undefined value (which is NOT the same thing as 0). This means the function is not defined at that point. However, just because a function is undefined at a point doesn't mean it doesn't have a limit. The limit is simply whichever value the function is getting close to.

One method of finding the limit is to try and simplify the equation as much as possible:

$f(x)=\frac{(x-3)(x+2)}{x+2}$

As you can see, there are common factors between the numerator and the denominator that can be canceled out. (Remember, when you cancel out a factor from a rational equation, it means that the function has a hole -- an undefined point -- where that factor equals zero.)

After canceling out the common factors, we're left with:

Even though the domain of the original function is restricted ( cannot equal ), we can still substitute into this simplified equation to find the limit at

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Question

Evaluate the limit below:

$y=\lim_{x \to\frac{\pi }2{} }(sinx-cosx)^{tanx}$

Tap to reveal answer

Answer

will approach $1^{\infty }$ when approaches $\frac{\pi }{2}$ , so $\lim_{x\rightarrow \frac{\pi }{2}} (lny)$ will be of type $0\cdot \infty$ as shown below:

$\lim_{x\rightarrow \frac{\pi }{2}} lny = \lim_{x\rightarrow \frac{\pi }{2}} ln (sinx-cosx)^{tanx}=\lim_{x\rightarrow \frac{\pi }{2}} tanx\cdot ln(sinx-cosx) = \infty \cdot 0$

So, we can apply the L’ Hospital's Rule:

$\lim_{x\rightarrow \frac{\pi }{2}} lny = \lim_{x\rightarrow \frac{\pi }{2}} \frac{sinx \cdot ln(sinx-cosx)}{cosx}=$

$\lim_{x\rightarrow \frac{\pi }{2}} \frac{cosx\cdot ln(sinx-cosx)+sinx\cdot (\frac{cosx+sinx}{sinx-cosx})}{-sinx}=\frac{0+1}{-1}=-1$

since:

$\lim_{x\rightarrow \frac{\pi }{2}} lny=-1$

hence:

$\lim_{x\rightarrow \frac{\pi }{2}} y=\lim_{x\rightarrow \frac{\pi }{2}} e^{lny}=e^{-1}=\frac{1}{e}$

← Didn't Know|Knew It →

Question

The speed of a car traveling on the highway is given by the following function of time:

What can you say about the car's speed after a long time (that is, as approaches infinity)?

Tap to reveal answer

Answer

The function given is a polynomial with a term , such that is greater than 1.

Whenever this is the case, we can say that the whole function diverges (approaches infinity) in the limit as approaches infinity.

This tells us that the given function is not a very realistic description of a car's speed for large !

← Didn't Know|Knew It →

Question

$\textup{Find: }\lim_{x \to \infty }\frac{2x^{2}-7x}{x^{2}+3}$

Tap to reveal answer

Answer

$\textup{As } x \to\infty,\textup{all but the terms with the highest exponents become insignificant.}$

$\textup{As }x^{2} \textup{ is the highest on top and bottom, evaluate the coefficients:}} \frac{2}{1}=2$

← Didn't Know|Knew It →

Question

Calculate $\lim_{x\rightarrow \infty }\left ( x^{2} \sin \frac{1}{ x^{2}-1} - \sin \frac{1}{ x^{2}-1} \right )$ .

Tap to reveal answer

Answer

This can be rewritten as follows:

$\lim_{x\rightarrow \infty }\left ( x^{2} \sin \frac{1}{ x^{2}-1} - \sin \frac{1}{ x^{2}-1} \right )$

$= \lim_{x\rightarrow \infty }\left ( x^{2}\cdot \sin \frac{1}{ x^{2}-1} -1\cdot \sin \frac{1}{ x^{2}-1} \right )$

$= \lim_{x\rightarrow \infty }\left [\left ( x^{2}-1 \right ) \cdot \sin \frac{1}{ x^{2}-1} \right]$

$= \lim_{x\rightarrow \infty } \frac{ \sin \frac{1}{ x^{2}-1} }{\frac{1}{ x^{2}-1}}$

We can substitute $u = \frac{1}{ x^{2}-1}$ , noting that as $x\rightarrow \infty$ , $u \rightarrow 0$ :

$= \lim_{u\rightarrow 0} \frac{ \sin u }{u} = 1$ , which is the correct choice.

← Didn't Know|Knew It →

Question

Let $f(x)=\frac{1}{x}$ .

Find $\lim_{x\ \to \0^{-}} \frac{1}{x}$ .

Tap to reveal answer

Answer

This is a graph of $\frac{1}{x}$ . We know that $\frac{1}{0}$ is undefined; therefore, there is no value for . But as we take a look at the graph, we can see that as approaches 0 from the left, approaches negative infinity.

This can be illustrated by thinking of small negative numbers.

NOTE: Pay attention to one-sided limit specifications, as it is easy to pick the wrong answer choice if you're not careful.

$\lim_{x \to \0^+} \frac{1}{x}$ is actually infinity, not negative infinity.

← Didn't Know|Knew It →

Question

$f(x)=\frac{x^{2}-x-6}{x+2}$

$\lim_{x\rightarrow -2}f(x) =$

Tap to reveal answer

Answer

A limit describes what -value a function approaches as approaches a certain value (in this case, ). The easiest way to find what -value a function approaches is to substitute the -value into the equation.

$f(-2)=\frac{(-2)^{2}-(-2)-6}{-2+2}=\frac{4+2-6}{0}=\frac{0}{0}$

Substituting for gives us an undefined value (which is NOT the same thing as 0). This means the function is not defined at that point. However, just because a function is undefined at a point doesn't mean it doesn't have a limit. The limit is simply whichever value the function is getting close to.

One method of finding the limit is to try and simplify the equation as much as possible:

$f(x)=\frac{(x-3)(x+2)}{x+2}$

As you can see, there are common factors between the numerator and the denominator that can be canceled out. (Remember, when you cancel out a factor from a rational equation, it means that the function has a hole -- an undefined point -- where that factor equals zero.)

After canceling out the common factors, we're left with:

Even though the domain of the original function is restricted ( cannot equal ), we can still substitute into this simplified equation to find the limit at

← Didn't Know|Knew It →

Question

Evaluate the limit below:

$y=\lim_{x \to\frac{\pi }2{} }(sinx-cosx)^{tanx}$

Tap to reveal answer

Answer

will approach $1^{\infty }$ when approaches $\frac{\pi }{2}$ , so $\lim_{x\rightarrow \frac{\pi }{2}} (lny)$ will be of type $0\cdot \infty$ as shown below:

$\lim_{x\rightarrow \frac{\pi }{2}} lny = \lim_{x\rightarrow \frac{\pi }{2}} ln (sinx-cosx)^{tanx}=\lim_{x\rightarrow \frac{\pi }{2}} tanx\cdot ln(sinx-cosx) = \infty \cdot 0$

So, we can apply the L’ Hospital's Rule:

$\lim_{x\rightarrow \frac{\pi }{2}} lny = \lim_{x\rightarrow \frac{\pi }{2}} \frac{sinx \cdot ln(sinx-cosx)}{cosx}=$

$\lim_{x\rightarrow \frac{\pi }{2}} \frac{cosx\cdot ln(sinx-cosx)+sinx\cdot (\frac{cosx+sinx}{sinx-cosx})}{-sinx}=\frac{0+1}{-1}=-1$

since:

$\lim_{x\rightarrow \frac{\pi }{2}} lny=-1$

hence:

$\lim_{x\rightarrow \frac{\pi }{2}} y=\lim_{x\rightarrow \frac{\pi }{2}} e^{lny}=e^{-1}=\frac{1}{e}$

← Didn't Know|Knew It →

Question

The speed of a car traveling on the highway is given by the following function of time:

What can you say about the car's speed after a long time (that is, as approaches infinity)?

Tap to reveal answer

Answer

The function given is a polynomial with a term , such that is greater than 1.

Whenever this is the case, we can say that the whole function diverges (approaches infinity) in the limit as approaches infinity.

This tells us that the given function is not a very realistic description of a car's speed for large !

← Didn't Know|Knew It →

Question

$\textup{Find: }\lim_{x \to \infty }\frac{2x^{2}-7x}{x^{2}+3}$

Tap to reveal answer

Answer

$\textup{As } x \to\infty,\textup{all but the terms with the highest exponents become insignificant.}$

$\textup{As }x^{2} \textup{ is the highest on top and bottom, evaluate the coefficients:}} \frac{2}{1}=2$

← Didn't Know|Knew It →

Question

Calculate $\lim_{x\rightarrow \infty }\left ( x^{2} \sin \frac{1}{ x^{2}-1} - \sin \frac{1}{ x^{2}-1} \right )$ .

Tap to reveal answer

Answer

This can be rewritten as follows:

$\lim_{x\rightarrow \infty }\left ( x^{2} \sin \frac{1}{ x^{2}-1} - \sin \frac{1}{ x^{2}-1} \right )$

$= \lim_{x\rightarrow \infty }\left ( x^{2}\cdot \sin \frac{1}{ x^{2}-1} -1\cdot \sin \frac{1}{ x^{2}-1} \right )$

$= \lim_{x\rightarrow \infty }\left [\left ( x^{2}-1 \right ) \cdot \sin \frac{1}{ x^{2}-1} \right]$

$= \lim_{x\rightarrow \infty } \frac{ \sin \frac{1}{ x^{2}-1} }{\frac{1}{ x^{2}-1}}$

We can substitute $u = \frac{1}{ x^{2}-1}$ , noting that as $x\rightarrow \infty$ , $u \rightarrow 0$ :

$= \lim_{u\rightarrow 0} \frac{ \sin u }{u} = 1$ , which is the correct choice.

← Didn't Know|Knew It →

Question

Let $f(x)=\frac{1}{x}$ .

Find $\lim_{x\ \to \0^{-}} \frac{1}{x}$ .

Tap to reveal answer

Answer

This is a graph of $\frac{1}{x}$ . We know that $\frac{1}{0}$ is undefined; therefore, there is no value for . But as we take a look at the graph, we can see that as approaches 0 from the left, approaches negative infinity.

This can be illustrated by thinking of small negative numbers.

NOTE: Pay attention to one-sided limit specifications, as it is easy to pick the wrong answer choice if you're not careful.

$\lim_{x \to \0^+} \frac{1}{x}$ is actually infinity, not negative infinity.

← Didn't Know|Knew It →

Question

$f(x)=\frac{x^{2}-x-6}{x+2}$

$\lim_{x\rightarrow -2}f(x) =$

Tap to reveal answer

Answer

A limit describes what -value a function approaches as approaches a certain value (in this case, ). The easiest way to find what -value a function approaches is to substitute the -value into the equation.

$f(-2)=\frac{(-2)^{2}-(-2)-6}{-2+2}=\frac{4+2-6}{0}=\frac{0}{0}$

Substituting for gives us an undefined value (which is NOT the same thing as 0). This means the function is not defined at that point. However, just because a function is undefined at a point doesn't mean it doesn't have a limit. The limit is simply whichever value the function is getting close to.

One method of finding the limit is to try and simplify the equation as much as possible:

$f(x)=\frac{(x-3)(x+2)}{x+2}$

As you can see, there are common factors between the numerator and the denominator that can be canceled out. (Remember, when you cancel out a factor from a rational equation, it means that the function has a hole -- an undefined point -- where that factor equals zero.)

After canceling out the common factors, we're left with:

Even though the domain of the original function is restricted ( cannot equal ), we can still substitute into this simplified equation to find the limit at

← Didn't Know|Knew It →

Question

Evaluate the limit below:

$y=\lim_{x \to\frac{\pi }2{} }(sinx-cosx)^{tanx}$

Tap to reveal answer

Answer

will approach $1^{\infty }$ when approaches $\frac{\pi }{2}$ , so $\lim_{x\rightarrow \frac{\pi }{2}} (lny)$ will be of type $0\cdot \infty$ as shown below:

$\lim_{x\rightarrow \frac{\pi }{2}} lny = \lim_{x\rightarrow \frac{\pi }{2}} ln (sinx-cosx)^{tanx}=\lim_{x\rightarrow \frac{\pi }{2}} tanx\cdot ln(sinx-cosx) = \infty \cdot 0$

So, we can apply the L’ Hospital's Rule:

$\lim_{x\rightarrow \frac{\pi }{2}} lny = \lim_{x\rightarrow \frac{\pi }{2}} \frac{sinx \cdot ln(sinx-cosx)}{cosx}=$

$\lim_{x\rightarrow \frac{\pi }{2}} \frac{cosx\cdot ln(sinx-cosx)+sinx\cdot (\frac{cosx+sinx}{sinx-cosx})}{-sinx}=\frac{0+1}{-1}=-1$

since:

$\lim_{x\rightarrow \frac{\pi }{2}} lny=-1$

hence:

$\lim_{x\rightarrow \frac{\pi }{2}} y=\lim_{x\rightarrow \frac{\pi }{2}} e^{lny}=e^{-1}=\frac{1}{e}$

← Didn't Know|Knew It →

Question

The speed of a car traveling on the highway is given by the following function of time:

What can you say about the car's speed after a long time (that is, as approaches infinity)?

Tap to reveal answer

Answer

The function given is a polynomial with a term , such that is greater than 1.

Whenever this is the case, we can say that the whole function diverges (approaches infinity) in the limit as approaches infinity.

This tells us that the given function is not a very realistic description of a car's speed for large !

← Didn't Know|Knew It →

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