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High School Math : Finding Limits as X Approaches Infinity

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Example Question #1 :Finding Limits As X Approaches Infinity

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

Possible Answers:

$\infty$

Correct answer:

Explanation:

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

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

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Example Question #1 :Finding Limits As X Approaches Infinity

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

Possible Answers:

$\infty$

The limit does not exist.

$\sin 1$

Correct answer:

Explanation:

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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Example Question #2 :Limits

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

v(t) = 5t^2+2t

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

Possible Answers:

The speed of the car depends on the starting speed.

The speed of the car approaches infinity.

The speed of the car approaches zero.

The speed of the car approaches a constant number.

Nothing can be concluded from the given function.

Correct answer:

The speed of the car approaches infinity.

Explanation:

The function given is a polynomial with a term t^n , such thatis greater than 1.

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

这告诉我们,给定的函数不是一个版本y realistic description of a car's speed for large!

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High School Math : Finding Limits as X Approaches Infinity

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 :Finding Limits As X Approaches Infinity

Example Question #1 :Finding Limits As X Approaches Infinity

Example Question #2 :Limits

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