Limits, Continuity & Graphs — practice questions

Evaluate $\displaystyle\lim_{x \to 5} \frac{x^2 - 25}{x - 5}$.

The function $g$ is defined by $g(x) = \dfrac{2x^2 - x - 6}{x - 2}$. State the domain of $g$, evaluate $\displaystyle\lim_{x\to 2} g(x)$, and describe the discontinuity at $x = 2$.

A function is defined by $f(x) = \begin{cases} ax + b & x < 1 \\ 3 & x = 1 \\ x^2 + ax & x > 1 \end{cases}$. Determine whether values of $a$ and $b$ exist that make $f$ continuous at $x = 1$. If so, find them; if not, prove no such values exist.

For $f(x) = x^3 - 6x^2 + 5$, find the intervals on which $f$ is decreasing.

For $f(x) = x^3 - 3x^2 - 9x$, find the intervals where $f$ is concave up and where it is concave down, and state the $x$-coordinate at which the concavity changes.

The function $f(x) = \begin{cases} ax^2 + bx & x \leq 2 \\ 3x - 2 & x > 2 \end{cases}$ is differentiable at $x = 2$. Find $a$ and $b$, and explain why differentiability imposes two conditions rather than one.