Math 162

Class 6

\(\displaystyle \int_0^1 \frac{e^x}{e^x + 1}\;dx\)

\(\displaystyle \int_0^1 \frac{e^x + 1}{e^x}\;dx\)

Hyperbolic Functions

  • Hyperbolic cosine: \[\cosh t = \frac{e^t + e^{-t}}{2}\]

  • Hyperbolic sine: \[\sinh t = \frac{e^t - e^{-t}}{2}\]

  • Hyperbolic tangent: \[\tanh t = \frac{\sinh t}{\cosh t} = \frac{e^t - e^{-1}}{e^t + e^{-t}}\]

Properties

Derivatives

Inverse Functions

Limits of Fractions

\[\lim_{x\to a} \frac{f(x)}{g(x)}\]

  • \(\displaystyle \lim_{x\to a} \frac{f(x)}{g(x)} = \frac{\lim_{x\to a} f(x)}{\lim_{x\to a}g(x)}\) if the limit of \(g\) is not zero.

  • If \(\displaystyle \lim_{x \to a}g(x) = 0\) and \(\displaystyle\lim_{x\to a}f(x) \neq 0\) then \(\displaystyle\lim_{x\to a} \frac{f(x)}{g(x)}\) does not exist.

  • The interesting case: \(\displaystyle\lim_{x\to a}f(x) = \lim_{x\to a}g(x) = 0\)

  • Another interesting case: none of the two limits exists. Much more complicated.

Type \(0/0\)

  • \(f(x)\) and \(g(x)\) have a common factor which is a power of \(x - a\).

    • \(\displaystyle \lim_{x\to 3}\frac{x^2 - 2x - 3}{x^2 - 5x + 6}\)

    • \(\displaystyle \lim_{x\to 4}\frac{\sqrt{x} - 2}{x - 4}\)

  • Sometimes there is no easy way to factor.

    • \(\displaystyle \lim_{x\to 1}\frac{\ln x}{x^2 - 1}\)

    • \(\displaystyle \lim_{x \to 0}\frac{\sin x}{2^x - 1}\)

\(\displaystyle \lim_{x\to 1}\frac{\ln x}{x^2 - 1}\)

x y ln x x 2 − 1 y = x − 1y =2( x − 1)

\(\displaystyle \lim_{x \to 0}\frac{\sin x}{2^x - 1}\)

x y sin x 2 x − 1 y = xy =(ln 2) x

\(\displaystyle \lim_{x \to \frac{\pi}{2}}\frac{\cos x}{\frac{x}{\pi} - \frac{1}{2}}\)

\(\displaystyle \lim_{x \to 0}\frac{\cos x - 1}{x^2}\)

l’Hospital’s Rule

If \(\displaystyle\lim_{x\to a}f(x) = \lim_{x\to a}g(x) = 0\) and if \[\lim_{x\to a}\frac{f'(x)}{g'(x)}\] exists, then \[\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}.\]

The same rule works if \(\displaystyle\lim_{x\to a}f(x)\) and \(\displaystyle\lim_{x\to a}g(x)\) are both infinite.

\(\displaystyle \lim_{x \to \frac{\pi}{2}}\frac{1 - \sin x}{1 + \cos 2x}\)

\(\displaystyle \lim_{x \to \infty }\frac{ x^2 }{ \exp(x) }\)

\(\displaystyle \lim_{x \to \infty }\frac{ \ln x }{ \sqrt{x} }\)

Indeterminate Forms

  • \(\frac{0}{0}\)

  • \(\frac{\infty}{\infty}\)

  • \(0\cdot \infty\)

  • \(\infty - \infty\)

  • \(0^0\)

  • \(\infty^0\)

  • \(1^\infty\)

\(\displaystyle \lim_{x \to 0^+ } x \ln x\)

\(\displaystyle \lim_{x \to \infty } x \exp x\)

\(\displaystyle \lim_{x \to -\infty } x \exp x\)

\(\displaystyle \lim_{x \to {\frac{\pi}{2}}^- } \left(\sec x - \tan x\right)\)

\(\displaystyle \lim_{x \to 0^+ } x^{\sin x}\)

\(\displaystyle \lim_{x \to \infty } x^{1/x}\)

\(\displaystyle \lim_{x \to 0^+ } (\cos x)^{1/x^2}\)