Horizontal asymptotes Determine limx →∞ f(x) and limx →-∞ f(x) for the following functions. Then

step 1 It is given that fx is equal to square root of x x to power 2 plus 1 divided by 2x plus 1. We ne

Horizontal asymptotes Determine limx →∞ f(x) and limx →-∞...

Horizontal asymptotes Determine $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for the following functions. Then give the horizontal asymptotes of $f(\text {if any})$.
$$f(x)=\frac{\sqrt{x^{2}+1}}{2 x+1}$$

Key Concepts

Behavior of Square Root Functions in Limits

When evaluating limits involving square root functions, it is important to understand how expressions like ?(x² + constant) approximate |x| for large values of x. This understanding is essential because the absolute value function behaves differently depending on whether x is positive or negative, affecting the evaluation of the limit.

Absolute Value Considerations

Because the square root of a squared term is the absolute value of the original term (i.e., ?(x²) = |x|), careful treatment is required when x is negative. This consideration influences the simplification of expressions in limits, ensuring that the sign of the variable is properly handled when determining the function's behavior at infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity. They represent the value that f(x) approaches as the input grows without bound, and are determined by finding the limits of the function as x approaches infinity and negative infinity.

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as x becomes very large in the positive or negative direction. This process typically involves simplifying the function to identify the dominant terms, which dictate the function's end behavior, and can lead to the recognition of horizontal asymptotes.

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