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Problem 38 Easy Difficulty

Use continuity to evaluate the limit.

$ \displaystyle \lim_{x \to 4} 3^{\sqrt{x^2 - 2x - 4}} $

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Video Transcript

This is a problem. Number 38 of the Stuart Calculus eighth Edition, section 2.5. Use continuity to evaluate the limit. The limited expertise for of the function. Three to the power of the square root function, um, of the quantity X squared minus two X minus four And we have our definition of continuity your and operate where if a function is this continues at a point. A. The limit as X approaches sing of dysfunction have is equal to the function, have evaluated a So let's determine the continuity of dysfunction Within the Scrooge function. We have a polynomial which is continuous on all real on the domain, about real numbers, the square root function as a domain associated with all values creator than or equal to zero. So if there's any exercises that make this negative within the square root, those would be not included in the domain for continuity. And then we have the exponential function here, three to this power, our which is continuous for all values of X. So the only real restriction is within the square root. However we see that we have X approaching for for gives us here a value for the polynomial, which is positive, so there will be no domain restrictions. We can be sure that as they purchased for this entire function is certainly continuous, and we can use our definition to evaluate this limit. Three. Race to the square root four. Squared It's two times four mine's for So there's going to be three to the square root 16, minus eight it is for which is three to the power square would afford, which is three squared gives us nine And we saw this limit using our definition of continuity The answer.