# How do you find the linearization at a=0 of #f(x) = e^(5 x)#?

Linearization uses the Linear Approximation Method.

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To find the linearization of ( f(x) = e^{5x} ) at ( a = 0 ), we first find the first derivative of ( f(x) ) with respect to ( x ), evaluate it at ( x = 0 ), and then use that value to form the equation of the tangent line at ( x = 0 ).

The derivative of ( f(x) = e^{5x} ) with respect to ( x ) is ( f'(x) = 5e^{5x} ).

Evaluate ( f'(x) ) at ( x = 0 ): ( f'(0) = 5e^{5(0)} = 5 ).

So, the slope of the tangent line at ( x = 0 ) is ( m = 5 ).

To find the y-intercept of the tangent line, we substitute ( x = 0 ) into ( f(x) ): ( f(0) = e^{5(0)} = 1 ).

Therefore, the equation of the tangent line at ( x = 0 ) is ( y = 5x + 1 ).

This is the linearization of ( f(x) = e^{5x} ) at ( a = 0 ).

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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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