1. Find \\[\\lim_{x \\to \\var{a}}(\\simplify[std]{{b}x+{c}})\\]
\nLimit = [[0]].
\n2.Find \\[\\lim_{x \\to \\var{a1}}(\\simplify[std]{{b1}x^2+{c1}x+{d1}})\\]
\nLimit = [[1]].
\n3. Find \\[\\lim_{x \\to \\var{a2}}\\left(\\simplify[std]{({b2}x+{c2})/({b3}x+{c3})}\\right)\\]
\nLimit = [[2]]
\nEnter all numbers as either integers or fractions but not as decimals.
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\n2. 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}}\\]
\n3. 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.
\nBut $\\simplify[]{{b3}*{a2}+{c3}={b3*a2+c3} }\\neq 0$ and so we can make the substitution safely.
\nSo \\[\\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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", "metadata": {"description": "Using simple substitution to find $\\lim_{x \\to a} bx+c$, $\\lim_{x \\to a} bx^2+cx+d$ and $\\displaystyle \\lim_{x \\to a} \\frac{bx+c}{dx+f}$ where $d\\times a+f \\neq 0$.
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