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Johnny Yi 6 years, 3 months ago

Created this as a copy of Blathnaid's copy of Find the limit of an algebraic fraction as parameter tends to a given value.

Name	Status	Author	Last Modified
Find the limit of an algebraic fraction as parameter tends to a given value	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Luis's copy of Find the limit of an algebraic fraction as parameter tends to a given value	draft	Luis Hernandez	30/11/2018 18:45
Blathnaid's copy of Find the limit of an algebraic fraction as parameter tends to a given value	draft	Blathnaid Sheridan	07/12/2018 08:47
Limit 1	draft	Johnny Yi	19/12/2018 01:29
Find the limit of an algebraic fraction as parameter tends to a given value	draft	Xiaodan Leng	10/07/2019 22:50

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Question statement

Let the function $f$ be given by $\displaystyle f(t)=\simplify{({a3} * t + {b3}) / ({a2} * t ^ 2 + {2 * b2 * a2} * t + {c2 + a2 * b2 ^ 2}) }$

Give any introductory information the student needs.

			Name	Type	Generated Value

w	number	6
stat1	number	4.6280329089
b3	integer	1
a3	integer	-3
a2	integer	1
b2	integer	-4
v	integer	-8
c2	integer	5
d2	integer	3
stat2	number	-3.9613662422

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xxxxxxxxxx
 
a2*d2^2+2*a2*b2*d2+c2+a2*b2^2

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Generated value: `number`

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Parts

Gap-fill
1. Gap Mathematical expression
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Find the following limit:

$\displaystyle \simplify{Limit(f(t),t,{d2}) }= \;$Unnamed gap.

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Advice

Graph of $f$.

{plotf(a2,b2,c2,a3,b3,stat1,stat2)}

Note that $\simplify{{a2} * t ^ 2 + {2 * b2 * a2} * t + {c2 + a2 * b2 ^ 2} ={a2}*(t+{b2})^2+{c2}} \gt 0$.

Hence the denominator of $f(t) \neq 0,\;\forall t \in \mathbb{R}$ and so $f$ is continuous at all points in $\mathbb{R}$.

This means that in part a) we can take the limit by simply subsituting $t=\var{d2}$ into the expression for $f(t)$ and we get:

\[\lim_{x \to \var{d2}}f(t)=\simplify[all,!otherNumbers,fractionNumbers,!collectNumbers]{({a3} * {d2} + {b3}) / ({a2} * {d2}^ 2 + {2 * b2 * a2} * {d2}+ {c2 + a2 * b2 ^ 2})={v}/{w} }\]

Similarly in part b) we have :

\[\lim_{x \to a}f(t)=\simplify[all,!collectNumbers,!otherNumbers,fractionNumbers]{({a3} * a + {b3}) / ({a2} * a^ 2 + {2 * b2 * a2} * a+ {c2 + a2 * b2 ^ 2})}\]

As noted above we can find this limit by simply putting $t=a$ into the formula for the function as $f$ is continuous at all points in $\mathbb{R}$.

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Taxonomy: mathcentre

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Johnny Yi 6 years, 3 months ago

Error in variable testing condition

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Generated value: `number`

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Limit 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Johnny Yi 6 years, 3 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`