Welcome To ...


AoMath: Limits With Algebra



Rules:
Advice 1: Try plugging in.
Advice 2: Try Factoring the numerator and denominator and cancelling factors. If this is successful go back to step one.
Advice 3: Multiply the top and bottom by the conjugate. (radicals)
Advice 4: Look for special limits.

Problem Limits 1: (advice 1 is needed and sufficient)
$$ \lim_{x \rightarrow 3} {x + 3} $$
$$ x = 3 \Longrightarrow 3 + 3 = \boxed{6} $$



Problem Limits 2: (advice 2 is needed and sufficient)
$$ \lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3} $$
$$ x = 3 \Longrightarrow \frac{3^2 - 9}{3 - 3} = 0/0 $$
$$ \lim_{x \rightarrow 3} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \rightarrow 3} (x + 3) = \boxed{6} $$

Problem Limits 3: (advice 3 is needed and sufficient)
$$ \lim_{x \rightarrow 25} \frac{x - 25}{\sqrt{x} - 5} $$
$$ x = 25 \Longrightarrow \frac{25 - 25}{5 - 5} = 0/0 $$
$$ \lim_{x \rightarrow 25} \frac{(x - 25)(\sqrt{x} + 5)}{(\sqrt{x} - 5)(\sqrt{x} + 5)} = \lim_{x \rightarrow 25} \frac{(x - 25)(\sqrt{x} + 5)}{x - 25} = \lim_{x \rightarrow 25} \sqrt{x} + 5 = 5 + 5 = \boxed{10} $$

Problem Limits 4: Special Limits
$$ \lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1 $$
$$ \lim_{x \rightarrow 0} \frac{x}{\sin(3x)} = ?? $$
$$ \lim_{x \rightarrow 0} \frac{3x}{3\sin(3x)} $$
$$ \lim_{x \rightarrow 0} \frac{1}{3}\frac{3x}{\sin(3x)} $$
$$ \lim_{x \rightarrow 0} \frac{1}{3}\frac{y}{\sin(y)} = \boxed{ \frac{1}{3} } $$
$$ \lim_{x \rightarrow 0} \frac{1 - \cos(x)}{x} = 0 $$
$$ \lim_{x \rightarrow 0} \frac{1 - \cos(0.2x)}{x} = ?? $$
$$ \lim_{x \rightarrow 0} \frac{0.2(1 - \cos(0.2x))}{0.2x} $$
$$ \lim_{x \rightarrow 0} 0.2(\frac{1-\cos(y)}{y}) $$
$$ \lim_{x \rightarrow 0} 0.2(0) = \boxed{0}$$
$$ \lim_{x \rightarrow \infty} \frac{1}{x} = 0 $$
$$ \lim_{x \rightarrow \infty} \frac{3x^3 - 5x^2 + 1}{11x^3 - 10} = ?? $$
$$ \lim_{x \rightarrow \infty} \frac{3x^3 - 5x^2 + 1}{11x^3 - 10} \frac{\frac{1}{x^3}}{\frac{1}{x^3}} $$
$$ \lim_{x \rightarrow \infty} \frac{3 - \frac{5}{x} + \frac{1}{x^3}}{11 - \frac{10}{x^3}} $$
all non-constant terms go to 0.

$$ \lim_{x \rightarrow \infty} \frac{3 + 0}{11 - 0} = \boxed{\frac{3}{11}} $$



Limits that do NOT exist


Use The Chart Method:
$$ \lim_{x \rightarrow 0} \frac{|x|}{x} = ?? $$

x = -1
x = -0.1
x = -0.01
x = -0.001
x = -0.0001
f(x)
-1
-1
-1
-1
\( \boxed{-1} \)

x = 1
x = 0.1
x = 0.01
x = 0.001
x = 0.0001
f(x)
1
1
1
1
\( \boxed{1} \)
$$ \lim_{x \rightarrow 0} \sin{(\frac{\pi}{2\cdot x})} = ?? $$
\( (x,f(x)) \)
\( (-1,\boxed{-1} \)
\( (-\frac{1}{2},\boxed{0}) \)
\( (-\frac{1}{3},\boxed{1}) \)
\( (-\frac{1}{4},\boxed{0}) \)
\( (-\frac{1}{5},\boxed{-1}) \)
\( (x,f(x)) \)
\( (1,\boxed{1}) \)
\( (\frac{1}{2},\boxed{0}) \)
\( (\frac{1}{3},\boxed{-1}) \)
\( (\frac{1}{4},\boxed{0}) \)
\( (\frac{1}{5},\boxed{1}) \)
It would seem neither of these last two limits exists as f(x) does not approach a single value.