Before start, limit of \(f\) when x-> \(c\) \(\neq f(c) \)
\( f(x) = \frac{x^{2}-1}{x-1} \) is defined everywhere except x=1
\( f(1) \) ?
\( f \) is not defined at \(x=1\)
When x close to 1, \(x\) -> \(1\)
f(x) close to 2, >>> \(f(x)\) -> \(2\)
\( \neq f(1) = 2 \)
Find limit from both left and right hand.
Definition of Limit
Left Limit
\(lim_{x \to c^{-}} f(x) = M\)
Example:
\( lim_{x \to 1^{-}} f(x) = 2 \)
Right Limit
\( lim_{x \to c^{+}} f(x) = N \)
Example: \( lim_{x \to 1^{+}} f(x) =2\)
Limit exists! When M=N
Existence of the Limit of a Function
\(f(1)=2\)
Left limit: \( lim_{x \to 1^{-}} f(x) = 2 \)
Right limit: \( lim_{x \to 1^{+}} f(x) = 2 \)
So, \( lim_{x \to 1} f(x) = 2 \)
\(g(1)=3\)
Left limit: \( lim_{x \to 1^{-}} g(x) = 2\)
Right limit: \( lim_{x \to 1^{+}} g(x) = 2 \)
So, \( lim_{x \to 1} g(x) = 2\)
\(y=f(x)\)
\(f(-2) = 2\) , just look at the black dot, there is no explanation for why “2”, this is how it works
Left limit: \( lim_{x \to -2^{-}} f(x) = 1 \)
Right limit: \( lim_{x \to -2^{+}} f(x) = 3\)
\( lim_{x \to -2} f(x) = \)does not exist as left \( lim \neq\) right \(lim\).
ONLY left limit equals to the right limit, limit exist.
Left limit when \( x \to 0^{-} \)
Keep trying values from the left.
| \( f(-0.1) \) | \( f(-0.01) \) | \( f(-0.001) \) | \( f(-0.0001) \) |
| \(= \frac{1}{-0.1} \) | \(= \frac{1}{-0.01} \) | \(= \frac{1}{-0.001} \) | \(= \frac{1}{-0.0001} \) |
| \(= -10 \) | \(= -100 \) | \(= -1000 \) | \(= -10000 \) |
\( \to lim_{x \to 0^{-}} f(x) = -\infty \)
Right limit when \( x \to 0^{+} \)
| \( f(0.0001) \) | \( f(0.001) \) | \( f(0.01) \) | \( f(0.1) \) |
| \(= \frac{1}{0.0001} \) | \(= \frac{1}{0.001} \) | \(= \frac{1}{0.01} \) | \(= \frac{1}{0.1} \) |
| 10000 | 1000 | 100 | 10 |
\( \to lim_{x \to 0^{+}} f(x) = +\infty \)
\( -\infty \neq +\infty \), therefore, limit does not exist.
Left limit when \( x \to 1^{-}\)
| \(x\) | \(0.9\) | \(0.99\) | \(0.999\) | \(0.9999\) |
| \( f(x)\) | \(1.9\) | \(1.99\) | \(1.999\) | \(1.9999\) |
\( \to lim_{x \to 1^{-}} f(x) = 2 \)
Right limit when \( x \to 1^{+}\)
| \(x\) | \(1.0001\) | \(1.001\) | \(1.01\) | \(1.1\) |
| \( f(x)\) | \(2.0001\) | \(2.001\) | \(2.01\) | \(2.1\) |
\( \to lim_{x \to 1^{+}} f(x) = 2 \)
\( lim_{x \to 1} f(x) =2 \)
We will not always check left lim and right lim in later, therefore, we have some tricks to check them.
\( lim_{x \to 1} \frac{(3x+2)(x-2)}{5x-2} = f(x) \)
not defined when \( x = \frac{2}{5} \)
Find limit:
\( \frac{[3(1)+2][1-2]}{5(1)-2} \)
\(= -\frac{5}{3} \)
Cannot directly sub 3 into x to check as it will become 0.
\( lim_{x \to 3} \frac{x^{2}-2x-3}{x-3} \)
\( = lim_{x \to 3} \frac{(x-3)(x+1)}{x-3} \)
\( = (3+1) \)
\( = 4 \)
It is nearly impossible to have \( \frac{0}{0} \) in any homework, quiz and exam, this is basically inviting you to factorise.
\( lim_{x \to 0} \frac{\sqrt{x+4}-2}{x} \)
\( = lim_{x \to 0} \frac{(\sqrt{x+4}-2)}{x} x \frac{(\sqrt{x+4}+2)}{\sqrt{x+4}+2} \)
\( = lim_{x \to 0} \frac{(x+4)-4}{x(\sqrt{x+4}+2)} \)
\( = lim_{x \to 0} \frac{x}{x(\sqrt{x+4}+2)} \)
\( = lim_{x \to 0} \frac{x}{x(\sqrt{x+4}+2)}\)
\( = lim_{x \to 0} \frac{1}{(\sqrt{x+4}+2)} \)
\( = \frac{1}{(\sqrt{0+4}+2)} = \frac{1}{4}\)
\( lim_{x \to \infty} \frac{1}{x} \) :it will get smaller and smaller, and 0 finally.
\( lim_{x \to \infty} \frac{1}{x} = 0 \)
\( f(x) = \frac{1}{x} \)
\( f(100) = \frac{1}{100} \)
\( f(1000) = \frac{1}{1000} \)
\(g(x)=x \)
\( lim_{x \to \infty} g(x) = +\infty \)
\(lim_{x \to +\infty} [\frac{1}{2^{x}} + \frac{1}{x^{2}+x}] \)
\(lim_{x \to +\infty}[\frac{1}{2^{x}}] + lim_{x \to +\infty}[\frac{1}{x^{2}+x}]\)
\(0+0=0\)
Example:
\( f(10) = \frac{1}{2^{10}} + \frac{1}{10^{2} + 10} \)
\( lim_{x \to \infty} [x^{4} – \pi x^{2}] \)
\( = lim_{x \to \infty} x^{4} – lim_{x \to \infty} \pi x^{2}\)
WRONG!!!
\( ~ lim_{x \to \infty} x^{4} = \infty\)
\( \to \) limit does not exist.
\( lim_{x \to \infty} \frac{2x^{3}+3}{3x^{3}+1} \)
Tips: always the guy with the highest power.
\( = lim_{x \to \infty} (\frac{2x^{3}+x}{3x^{3}+1}) (\frac{\frac{1}{x^{3}}}{\frac{1}{x^{3}}}) \)
— As of 2023-09-19 19:30
___ Start from 2023-09-26 16:50:23
Slope of any function = derivative: \( \frac{df}{dx} = f(x) = y’ \)
Differentiation Rules:
- \( \frac{d}{dx}(C) = 0\)
- \( \frac{d}{dx}(x) = 1\)
- \( \frac{d}{dx}(x^{n})=nx^{n-1} \)
- \( \frac{d}{dx}[u(x)+v(x)] = \frac{d}{dx}u(x) + \frac{d}{dx}v(x) \)
- \( \frac{d}{dx}[Cu(x)] = C \frac{d}{dx}u(x)\), C is constant.
