Limits and Continuity

A student is trying to solve this problem $\lim_{x \to 0} \frac{\sin 5x}{x}$. Here is what this student did Step 1: $\lim_{x \to 0} \frac{\sin 5x}{x} \left( \frac{5}{5} \right)$ Step 2: $\lim_{x \to 0} \frac{5 \sin 5x}{5x}$ Step 3: $5 \cdot 1 = 5$ Is this student correct?

What is $\lim_{x \to -4} \frac{x^2 - x - 20}{x + 4}$?

Given that $\lim_{{x \to -\infty}} g(x) = L$ and $\lim_{{h \to 0}} \frac{g(-1+h)-g(-1)}{h} = m$, which expression accurately expresses L in terms of m?

Find the limit $(x \to 2)(3x^2 - 2x + 5)$.

For what value of k does the function f(x) = \left{\begin{array}{ll} k^3 - k & \text{if } x < a \\ kx & \text{if } x \geq a \end{array}\right} have a continuous limit at $x=a$?

True or False: the limit of a constant a constant.

What type of discontinuity does $h(p)=\sqrt{\left|p\right|-4}/(\left|p\right|-4)$ exhibit as $p$ approaches $\pm 4$?

A pair of students are trying to solve this problem: $\lim_{x \to 0} \sec(2x)$. The following are the steps the students follow: Step 1: $\lim_{x \to 0} \sec(2x) = \lim_{x \to 0} \sec \cdot \lim_{x \to 0} (2x)$ Step 2 = $0 \cdot 0$ Step 3 = $0$ Are the students correct?

If $h(x) = \frac{f(x)}{x^2}$ and $\lim_{x \to 0} f(x) = 0$, what is the limit of $h(x)$ as $x$ approaches 0 given that $f'(x)$ exists and is continuous near 0?

Find the limit of the function $f(x) = \frac{3x^2 - 4x + 2}{2x^2 - 5x + 3}$ as $x$ approaches 1.