For #f(x)=(2x+1)/(x+2) # what is the equation of the tangent line at #x=1#?
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First you need to find the slope
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The equation of a line passing through
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The equation of the tangent line at x=1 for the function f(x)=(2x+1)/(x+2) is y=3x1.
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To find the equation of the tangent line at ( x = 1 ) for the function ( f(x) = \frac{2x + 1}{x + 2} ), we need to find the slope of the tangent line at ( x = 1 ) and then use the pointslope form of a line.

Find the derivative of the function: [ f'(x) = \frac{d}{dx}\left(\frac{2x + 1}{x + 2}\right) ] Using the quotient rule: [ f'(x) = \frac{(2)(x + 2)  (2x + 1)(1)}{(x + 2)^2} ] [ f'(x) = \frac{2x + 4  2x  1}{(x + 2)^2} ] [ f'(x) = \frac{3}{(x + 2)^2} ]

Evaluate the derivative at ( x = 1 ) to find the slope of the tangent line: [ f'(1) = \frac{3}{(1 + 2)^2} = \frac{3}{9} = \frac{1}{3} ]

Find the value of ( f(1) ): [ f(1) = \frac{2(1) + 1}{1 + 2} = \frac{3}{3} = 1 ]

Now, we have the slope (( m = \frac{1}{3} )) and a point (( x = 1, y = 1 )) on the tangent line. Use the pointslope form of a line to find the equation of the tangent line: [ y  y_1 = m(x  x_1) ] [ y  1 = \frac{1}{3}(x  1) ] [ y  1 = \frac{1}{3}x  \frac{1}{3} ] [ y = \frac{1}{3}x + \frac{2}{3} ]
So, the equation of the tangent line at ( x = 1 ) is ( y = \frac{1}{3}x + \frac{2}{3} ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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