Show that f is continuous at a. f(x)=√(2 x-5)+3 x ; a=4 |...

Name: Problem 19, Chapter 2: Calculus
Uploaded: 2022-03-05T20:53:31-08:00
Duration: 4 min 32 s
Channel: Eleni Katirtzoglou
Description: Show that f is continuous at a. f(x)=√(2 x-5)+3 x ; a=4

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Key Concepts

Continuity at a Point

A function is continuous at a specific point if the limit of the function as x approaches that point equals the function’s value at that point. This concept is fundamental in calculus as it ensures that small changes in x lead to small changes in f(x), making the function predictable around that point.

Limit Laws

Limit laws allow us to break down the limit of a combination of functions into the limits of individual functions. This is important for evaluating the overall limit of a function defined by sums, products, and compositions of simpler functions, provided each individual limit exists.

Direct Substitution

When dealing with continuous functions, the limit as x approaches a certain value can often be computed by simply substituting that value into the function. This method relies on the function being well-behaved (i.e., continuous) at the point of interest.

Domain Considerations

In functions involving operations like square roots, it is essential to ensure that the expression under the square root is within the domain (non-negative). Checking that the point of continuity lies within the domain is a necessary part of establishing continuity.

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