Difference between revisions of "1950 AHSME Problems/Problem 40"
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==Problem== | ==Problem== | ||
| − | The limit of <math> \frac {x^2 | + | The limit of <math> \frac {x^2-1}{x-1}</math> as <math>x</math> approaches <math>1</math> as a limit is: |
<math>\textbf{(A)}\ 0 \qquad | <math>\textbf{(A)}\ 0 \qquad | ||
| Line 8: | Line 8: | ||
\textbf{(D)}\ 2 \qquad | \textbf{(D)}\ 2 \qquad | ||
\textbf{(E)}\ 1</math> | \textbf{(E)}\ 1</math> | ||
| + | |||
| + | ==Solution== | ||
| + | Both <math>x^2-1</math> and <math>x-1</math> approach 0 as <math>x</math> approaches <math>1</math>, using the L'Hôpital's rule, we have <math>\lim \limits_{x\to 1}\frac{x^2-1}{x-1} = \lim \limits_{x\to 1}\frac{2x}{1} = 2</math>. | ||
| + | Thus, the answer is <math>\boxed{\textbf{(D)}\ 2}</math>. | ||
| + | |||
| + | ~ MATH__is__FUN | ||
| + | |||
| + | ==Solution 2== | ||
| + | The numerator of <math> \frac {x^2-1}{x-1}</math> can be factored as <math>(x+1)(x-1)</math>. The <math>x-1</math> terms in the numerator and denominator cancel, so the expression is equal to <math>x+1</math> so long as <math>x</math> does not equal <math>1</math>. Looking at the function's behavior near 1, we see that as <math>x</math> approaches one, the expression approaches <math>\boxed{\textbf{(D)}\ 2}</math>. | ||
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| + | |||
| + | Note: Alternatively, we can ignore the domain restriction and just plug in <math>x = 1</math> into the reduced expression. | ||
| + | |||
| + | ~ [https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi] | ||
| + | |||
| + | {{AHSME 50p box|year=1950|num-b=39|num-a=41}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 12:43, 4 April 2024
Problem
The limit of 
as 
approaches 
as a limit is:
Solution
Both 
and 
approach 0 as approaches , using the L'Hôpital's rule, we have 
.
Thus, the answer is 
.
~ MATH__is__FUN
Solution 2
The numerator of can be factored as 
. The terms in the numerator and denominator cancel, so the expression is equal to 
so long as does not equal . Looking at the function's behavior near 1, we see that as approaches one, the expression approaches .
Note: Alternatively, we can ignore the domain restriction and just plug in 
into the reduced expression.
~ cxsmi
| 1950 AHSC (Problems • Answer Key • Resources) | ||
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Followed by Problem 41 | |
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