Are the functions continuous at the point being approached? limx →0^+ sin((π)/(2) e^√(x))

Name: Problem 37, Chapter 2: University Calculus: Early Transcendentals
Uploaded: 2021-08-03T03:50:52-08:00
Duration: 1 min 4 s
Channel: Carson Merrill
Description: Are the functions continuous at the point being approached? limx →0^+ sin((π)/(2) e^√(x))

step 1 For the given problem, we want to determine if the functions are continuous as x approaches 0 fr

Are the functions continuous at the point being approached?...

Key Concepts

Composition of Functions

Function composition involves applying one function to the result of another. When dealing with limits and continuity, if the outer function is continuous at the limit of the inner function, the overall limit can be evaluated by composing the limits, making this method a powerful tool in calculus.

Sine Function

The sine function is a trigonometric function that is continuous and periodic. Its continuity means that when it is composed with another function whose limit exists, the limit of the composition can be obtained by taking the sine of the limit of the inner function.

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as its argument approaches a particular value. They provide a way to analyze the function's behavior near points where it might not be explicitly defined or where its behavior changes, thereby serving as the foundation for defining continuity and derivatives.

Continuity

Continuity is a property of functions where the function's value at a point is consistent with the limit of the function as its argument approaches that point. A function is continuous at a point if the limit exists and equals the function’s value at that point, ensuring there are no abrupt changes in the function's graph.

Exponential Function

The exponential function, especially in contexts where it is composed with other functions like roots, is key in analyzing growth or decay behaviors. Its properties, such as always being positive and smoothly varying, make it an important candidate for substitution in limit evaluations when composed with other continuous functions.

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