# Which of the twelve basic functions are bounded above?

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The twelve basic functions are typically considered to be:

- Constant function: ( f(x) = c )
- Identity function: ( f(x) = x )
- Squaring function: ( f(x) = x^2 )
- Cubing function: ( f(x) = x^3 )
- Square root function: ( f(x) = \sqrt{x} )
- Cube root function: ( f(x) = \sqrt[3]{x} )
- Reciprocal function: ( f(x) = \frac{1}{x} )
- Exponential function: ( f(x) = e^x )
- Natural logarithm function: ( f(x) = \ln(x) )
- Sine function: ( f(x) = \sin(x) )
- Cosine function: ( f(x) = \cos(x) )
- Tangent function: ( f(x) = \tan(x) )

Among these functions, the bounded above functions are:

- Constant function: ( f(x) = c ) - Bounded above if ( c ) is finite.
- Squaring function: ( f(x) = x^2 ) - Not bounded above.
- Cubing function: ( f(x) = x^3 ) - Not bounded above.
- Square root function: ( f(x) = \sqrt{x} ) - Bounded above.
- Cube root function: ( f(x) = \sqrt[3]{x} ) - Bounded above.
- Reciprocal function: ( f(x) = \frac{1}{x} ) - Not bounded above.
- Exponential function: ( f(x) = e^x ) - Bounded above.
- Natural logarithm function: ( f(x) = \ln(x) ) - Not bounded above.
- Sine function: ( f(x) = \sin(x) ) - Bounded above.
- Cosine function: ( f(x) = \cos(x) ) - Bounded above.
- Tangent function: ( f(x) = \tan(x) ) - Not bounded above.

So, the functions bounded above among the twelve basic functions are the constant function, square root function, cube root function, exponential function, and sine function.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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