Continuity and differentiability are fundamental concepts in calculus that describe the behavior of functions.
A function is considered continuous at a point if there are no breaks, jumps, or holes in its graph at that point. Informally, we can think of a continuous function as one that can be drawn without lifting the pencil from the paper.
The function $f(x) = x^2$ is continuous for all real numbers. On the other hand, the function $g(x) = 1/x$ is continuous everywhere except at $x = 0$, where it has a vertical asymptote.
A function is differentiable at a point if it has a well-defined derivative at that point. Geometrically, this means the function has a unique tangent line at that point.
All differentiable functions are continuous, but not all continuous functions are differentiable.
The function $f(x) = |x|$ (absolute value of x) is continuous everywhere but not differentiable at $x = 0$ because it has a sharp corner at this point.
Continuity and differentiability are not tested in the exam.
Limits are a crucial concept in calculus, forming the foundation for understanding continuity, derivatives, and integrals.
When we talk about limits, we're often interested in what happens to a function as its input approaches a particular value. This behavior can be classified as either convergent or divergent.
Consider the limit:
$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$
This limit converges to 4, which we can see by factoring the numerator:
$$\lim_{x \to 2} \frac{(x+2)(x-2)}{x - 2} = \lim_{x \to 2} (x+2) = 4$$
On the other hand, the limit:
$$\lim_{x \to 0} \frac{1}{x}$$
diverges because the function approaches positive infinity as $x$ approaches 0 from the right, and negative infinity as $x$ approaches 0 from the left.
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