What does the graph of $x\cos(\frac{1}{x})$ squeezed between $-|x|$ and $|x|$ tell us?

It visually confirms that as $x$ approaches 0, $x\cos(\frac{1}{x})$ is forced to approach 0 as well.

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What does the graph of $x\cos(\frac{1}{x})$ squeezed between $-|x|$ and $|x|$ tell us?

It visually confirms that as $x$ approaches 0, $x\cos(\frac{1}{x})$ is forced to approach 0 as well.

What is the Squeeze Theorem?

If $f(x) leq g(x) leq h(x)$ and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$ , then $\lim_{x \to a} g(x) = L$ .

Define 'limit' in calculus.

The value that a function approaches as the input approaches a specific value.

What are bounding functions?

Functions that enclose another function, used in the Squeeze Theorem.

Define continuity at a point.

A function $f(x)$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$ .

State the Squeeze Theorem.

If $f(x) \leq g(x) \leq h(x)$ for all $x$ in an interval containing $a$ (except possibly at $a$ itself) and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$ , then $\lim_{x \to a} g(x) = L$ .

All Flashcards

What does the graph of $x\cos(\frac{1}{x})$ squeezed between $-|x|$ and $|x|$ tell us?

It visually confirms that as $x$ approaches 0, $x\cos(\frac{1}{x})$ is forced to approach 0 as well.

What is the Squeeze Theorem?

If $f(x) leq g(x) leq h(x)$ and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$ , then $\lim_{x \to a} g(x) = L$ .

Define 'limit' in calculus.

The value that a function approaches as the input approaches a specific value.

What are bounding functions?

Functions that enclose another function, used in the Squeeze Theorem.

Define continuity at a point.

A function $f(x)$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$ .

State the Squeeze Theorem.

If $f(x) \leq g(x) \leq h(x)$ for all $x$ in an interval containing $a$ (except possibly at $a$ itself) and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$ , then $\lim_{x \to a} g(x) = L$ .