Jeanne's copy of Ida Friestad's copy of Finding limits by substitution,

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Question 1

Find the following limits.

1. Find $\lim_{x \to \var{a}}(\simplify[std]{{b}x+{c}})$

Limit = Expected answer:.

2.Find $\lim_{x \to \var{a1}}(\simplify[std]{{b1}x^2+{c1}x+{d1}})$

Limit = Expected answer:.

3. Find $\lim_{x \to \var{a2}}\left(\simplify[std]{({b2}x+{c2})/({b3}x+{c3})}\right)$

Limit = interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{-\frac{ 28 }{ 53 }}$

Enter all numbers as either integers or fractions but not as decimals.

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Advice

1. To find this limit we simply substitute $x=\var{a}$ into $\simplify[std]{{b}x+{c}}$ to get $\lim_{x \to \var{a}}(\simplify[std]{{b}x+{c}})=\simplify[]{{b}*{a}+{c}={ans1}}$

2. Similarly to find this limit we simply substitute $x=\var{a1}$ into $\simplify[std]{{b1}x^2+{c1}x+{d1}}$ to get $\lim_{x \to \var{a1}}(\simplify[std]{{b1}x^2+{c1}x+{d1}}) =\simplify[]{{b1}*{a1}^2+{c1}*{a1}+{d1}={ans2}}$

3. Once again we could simply substitute $x=\var{a2}$ into $\displaystyle \simplify[std]{({b2}x+{c2})/({b3}x+{c3})}$ . However before doing this we must make sure that the denominator is not $0$ as otherwise we have a problem and the limit may not exist.

But $\simplify[]{{b3}*{a2}+{c3}={b3*a2+c3} }\neq 0$ and so we can make the substitution safely.

So $\lim_{x \to \var{a2}}\left(\simplify[std]{({b2}x+{c2})/({b3}x+{c3})}\right)=\simplify[]{({b2}*{a2}+{c2})/({b3}*{a2}+{c3})}=\simplify[all,fractionNumbers]{{b2*a2+c2}/{b3*a2+c3}}$

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Jeanne's copy of Ida Friestad's copy of Finding limits by substitution,

Advice