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High School Math : Finding Limits as X Approaches Infinity

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Example Question #2 : Limits

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

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 #2 : 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:

The limit does not exist.

$\infty$

$\sin 1$

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 #4 : Find The Limit Of A Function

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, as approaches infinity)?

Possible Answers:

The speed of the car depends on the starting speed.

The speed of the car approaches a constant number.

The speed of the car approaches infinity.

Nothing can be concluded from the given function.

The speed of the car approaches zero.

Correct answer:

The speed of the car approaches infinity.

Explanation:

The function given is a polynomial with a term t^n , 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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High School Math : Finding Limits as X Approaches Infinity

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #2 : Limits

Example Question #2 : Finding Limits As X Approaches Infinity

Example Question #4 : Find The Limit Of A Function

All High School Math Resources

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