How do you use the limit definition to find the slope of the tangent line to the graph #f(x)=sqrt(x+2)# at x=2?
To solve this limit we may "razionalize" the expression:
The limit becomes
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To find the slope of the tangent line to the graph of f(x) = √(x+2) at x=2 using the limit definition, we can follow these steps:

Start with the equation of the function: f(x) = √(x+2).

Determine the derivative of the function f(x) using the limit definition. The derivative, denoted as f'(x), represents the slope of the tangent line at any given point.

Apply the limit definition of the derivative to find f'(x). The limit definition states that f'(x) = lim(h→0) [f(x+h)  f(x)] / h.

Substitute the given value of x=2 into the limit definition equation: f'(2) = lim(h→0) [f(2+h)  f(2)] / h.

Evaluate the limit as h approaches 0. This involves plugging in the value of h=0 into the equation obtained in step 4 and simplifying the expression.

Simplify the expression and calculate the limit to find the slope of the tangent line at x=2.
By following these steps, you can use the limit definition to find the slope of the tangent line to the graph of f(x) = √(x+2) at x=2.
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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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