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Use continuity to evaluate the limit.

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

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Missouri State University

Harvey Mudd College

Baylor University

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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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.

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