Question

Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for $f(x).$ $$\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{2}+5}}{2 x}$$

Name: Problem 77, Chapter 3: Calculus with Applications
Uploaded: 2020-01-27T18:58:57-08:00
Duration: 57 s
Channel: Julian Wong
Description: Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f(x). limx →∞ (√(9 x^2+5))/(2 x)

   Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for $f(x).$
$$\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{2}+5}}{2 x}$$

Calculus with Applications

Margaret L. Lial •… 11th Edition

Chapter 3, Problem 77

Instant Answer

Solved by Expert Julian Wong

01/27/2020

Step 1

However, because the $x$ is inside a square root, we divide by $x^2$ instead. This gives us: $$\lim _{x \rightarrow \infty} \frac{\sqrt{9 + \frac{5}{x^2}}}{2}$$ Show more…

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Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for $f(x).$ $$\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{2}+5}}{2 x}$$

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Key Concepts

Limit at Infinity

This concept involves determining the behavior of a function as the independent variable approaches infinity. It focuses on identifying the dominant terms in the function's expression to ascertain what value the function approaches at extreme values of the variable.

Graphical Analysis of Functions

Using a graphing calculator to study a function's behavior allows one to visually confirm the limit at infinity. By analyzing the graph, one can observe how the function levels off, which provides an empirical method to estimate the end behavior of the function.

Algebraic Simplification in Limit Evaluation

This technique involves manipulating the expression algebraically, such as by dividing the numerator and the denominator by the highest power of the variable present. This process simplifies the function and makes it easier to identify the limit by reducing the function to its dominant form.

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