HS20

44/48 points.

MC

Prove a Limit does not Exist

Sequence criterion: we show that for two different series $(x_n),(y_n)$ the limits don’t agree ${\lim }_{n}f(x_n)\ne {\lim }_{n}f(y_n)$.

Other standard tools:

• one-sided limits differ → easy
• ${\lim }_{x\to a}$: find a sequence in a neighbourhood of $a$ that diverges.

How to choose:

• $f$ oscillates → pick sequences that hits peaks / throughs
• $f$ has jump → come from each side

Example ${\lim }_{x\to 0}\frac{1+\pi \sin (x)-e^x}{\tan (x^2)}$

• Both $\to 0$ so we can apply l’Hôpital’s
• we get $\frac{{\cos }^2(x^2)(\pi \cos (x)-e^x)}{2x}$
• for $\to 0^{+}$ this tends to $+\infty$
• for $\to 0^{-}$ this tends to $-\infty$
• Thus there is no limit

Improper Limits

Does this diverge ${\int }_1^{\infty }\frac{\sin (x)+1}{x}dx$?

The 1/x term contributes $\ln (R)$ for the integral from $1\to R$, with $R\to \infty$.
This term diverges.

$\int \sin (x)dx=-\cos (x)$ stays bounded, so it doesn’t actually do much.
Thus $1/x$ dominates $\sin (x)/x$ and as $1/x\to 0$, we get a convergent term.

divergent + convergent = divergent. done.