Revision sheet

The idea of a limit

• $\displaystyle\lim_{x \to a} f(x)$: where $f(x)$ heads as $x$ approaches $a$.
• $f(a)$ itself does not matter (it may not even exist).
• $\dfrac{0}{0}$ means: simplify, then look again.

Watch out: the limit is not always $f(a)$.

Limits with algebra

1. Substitute. A number? That is the limit.
2. $\dfrac{0}{0}$? Factor and cancel, expand, or use conjugates.
3. Substitute again.

Watch out: $\dfrac{0}{0}$ is never the answer.

One-sided limits

• $x \to a^-$: from the left; $x \to a^+$: from the right.
• The limit exists only if both sides agree.

Watch out: one side alone is not the two-sided limit.

Special limits

• $\dfrac{\sin u}{u} \to 1$ as $u \to 0$ (radians); so $\dfrac{\sin(ax)}{x} \to a$.
• Squeeze: trapped between two functions with the same limit.

Watch out: $\dfrac{\sin(ax)}{x}$ tends to $a$, not 1.

Limits at infinity

• Divide by the highest power in the denominator.
• Top degree smaller: 0. Equal: ratio of leading coefficients. Larger: $\pm\infty$.
• A finite limit $L$ gives the asymptote $y = L$.

Watch out: constants do not decide the limit at infinity.

Infinite limits

• Denominator → 0, numerator not: $\pm\infty$, a vertical asymptote.
• Test a value just beside $a$ for the sign.

Watch out: left and right of the asymptote, the signs can differ.

Continuity

1. $f(a)$ exists.
2. $\displaystyle\lim_{x \to a} f(x)$ exists.
3. They are equal.

Watch out: a limit that exists is not enough; $f(a)$ must match it.

The Intermediate Value Theorem

1. Check that $f$ is continuous on $[a, b]$.
2. Compare $f(a)$ and $f(b)$ with $N$ (often 0).
3. If $N$ is between them, $f(c) = N$ for some $c$ in $(a, b)$.

Watch out: no continuity, no conclusion.