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# Limits of Functions

What happens when a function's output isn't calculable directly – e.g., at infinity – but we still need to understand its behavior? That's where limits come in.

\[\lim_{x \to \infty}x^2\]

If the values of \(x\) increase without bound, what do the values of \(x^2\) approach?

\[A = \lim_{x \to 3}x-9\]

\[B = \lim_{x \to 3}x^2-9\]

\[C = \lim_{x \to 3}x^3-9\]

Which limit is equal to \(0?\)

\[A = \lim_{x \to 0}\frac{1}{x}\]

\[B = \lim_{x \to 0}\frac{x}{x}\]

\[C = \lim_{x \to 0}\frac{x^2}{x}\]

Which limit is equal to \(0?\)

\[\lim_{x \to 0}\frac{x^3}{x^m} = 1\]

What must be true of \(m\)?

\[\lim_{x \to 5}\frac{x^2-10x+25}{\left(x-5\right)^2}\]

If the value of \(x\) approaches 5, what does the value of \(\frac{x^2-10x+25}{\left(x-5\right)^2}\) approach?

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