Answer:
The tangent line is: y = 8x - 58
The tangent point is at (7, -2)
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Step-by-step explanation:
One of the definitions of derivatives is

where f ' (a) represents the derivative evaluated at x = a.
The value of f ' (a) will get us the slope of the tangent at x = a.
The idea is that x is getting closer and closer to 'a'. In doing so, the secant lines slowly approach the tangent line.
Keep in mind that x will never reach 'a' itself (if it did, then we'd have a division by zero error).
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The given limit we have is

and that is equivalent to

and also equivalent to

Compare that to the template I mentioned at the top to see that
- a = 7
- f(a) = f(7) = -2
- f ' (a) = f ' (7) = 8
Therefore, we can say the tangent slope is 8 and the tangent touches the f(x) curve at (x,y) = (a, f(a)) = (7,f(7)) = (7,-2)
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Let's use that info to determine the equation of the tangent line.
I'll use point-slope form to isolate y.

That's the equation of the tangent line to the point (7,-2).