True or False? The limit of a function at a point depends on the function's value at that point.

False. The limit is concerned with the function's behaviour as it approaches the point, not its value at the point.

What is a continuous function ?

A continuous function is a function whose graph can be drawn without lifting the pencil from the paper, with no holes or sudden jumps.

What does it mean for a function to be differentiable at a point?

A function is differentiable at a point if its derivative exists and has a well-defined value at that point.

True or False? A function can be differentiable at a point even if it is not continuous at that point.

False. For a function to be differentiable at a point, it must be continuous at that point.

True or False? If a function is continuous at a point, it is always differentiable at that point.

False. A function can be continuous at a point without being differentiable at that point. 'Differentiable at a point' implies 'continuous at a point', but 'continuous at a point' doesn't necessarily imply 'differentiable at a point'.

What does it mean for a function to be ' smooth ' at a point?

A function is smooth at a point if its graph does not have any corners or sudden changes of direction at that point.

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Basic Limits & Continuity (DP IB Analysis & Approaches (AA))

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What is a limit in mathematics?

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What is a limit in mathematics?
A limit in mathematics is the tendency of a mathematical process as it approaches, but never quite reaches, an 'end point' of some sort.
E.g. the limit of $\frac{1}{x}$ as $x$ gets bigger and bigger is zero, because $\frac{1}{x}$ tends to zero as $x$ tends to infinity.
What notation is used to denote a limit?
The notation $\lim_{x \to a} f (x)$ denotes "the limit of the function $f (x)$ as $x$ goes to (or approaches) a".
True or False?
The limit of a function at a point depends on the function's value at that point.
False.
The limit is concerned with the function's behaviour as it approaches the point, not its value at the point.
What does $\lim_{x \to \infty} f (x)$ represent?
$\lim_{x \to \infty} f (x)$ represents the limit of the function $f (x)$ as $x$ gets infinitely big in the positive direction
What is a continuous function?
A continuous function is a function whose graph can be drawn without lifting the pencil from the paper, with no holes or sudden jumps.
What does it mean for a function to be differentiable at a point?
A function is differentiable at a point if its derivative exists and has a well-defined value at that point.
True or False?
A function can be differentiable at a point even if it is not continuous at that point.
False.
For a function to be differentiable at a point, it must be continuous at that point.
True or False?
If a function is continuous at a point, it is always differentiable at that point.
False.
A function can be continuous at a point without being differentiable at that point.
'Differentiable at a point' implies 'continuous at a point', but 'continuous at a point' doesn't necessarily imply 'differentiable at a point'.
What does it mean for a function to be 'smooth' at a point?
A function is smooth at a point if its graph does not have any corners or sudden changes of direction at that point.
What is the limit of $\frac{k}{x^{n}}$ as $x$ approaches $\pm \infty$ , for $n > 0$ and $k \in ℝ$ ?
The limit of $\frac{k}{x^{n}}$ as $x$ approaches $\pm \infty$ , for $n > 0$ and $k \in ℝ$ , is zero.
True or False?
$\lim_{x \to \infty} e^{- p x} = 1$ for any $p > 0$ .
False.
$\lim_{x \to \infty} e^{- p x} = 0$ for any $p > 0$ .

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