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Chapter 1: Problem 8

Estimating a Limit Numerically In Exercises \(5-10\) , complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$\lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x}$$

Short Answer

Expert verified
The limit of the function as \(x\) approaches 0 is \(0.5\).

Step by step solution

01

Rationalize the Numerator

Multiply the numerator and denominator by the conjugate of \(\sqrt{x+1}-1\), which is \(\sqrt{x+1}+1\), in order to rationalize the numerator. Doing so results in the expression: \(\frac{(\sqrt{x+1}-1)(\sqrt{x+1}+1)}{x(\sqrt{x+1}+1)}\).
02

Simplify the Expression

Simplify the expression \((\sqrt{x+1}-1)(\sqrt{x+1}+1)\) into \(x+1-1\), which equals \(x\). The new function becomes \(\frac{x}{x(\sqrt{x+1}+1)}.\)
03

Further Simplification

Divide \(x\) by \(x\) in the new function to simplify the numerator to \(1\), resulting in: \(\frac{1}{\sqrt{x+1}+1}\).
04

Evaluate the Limit

Finally, plug in \(x = 0\) into the simplified function \(\frac{1}{\sqrt{x+1}+1}\) and we get that the limit is \(\frac{1}{\sqrt{1}+1} = \frac{1}{2}\).
05

Graph and Confirm

Graph the original function and verify that it approaches the limit as x approaches 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerical Estimation
Numerical estimation in calculus involves predicting the limit of a function as it approaches a specific point through calculated approximations. For our example, we first use a table of values. As we choose values of \(x\) that get increasingly close to zero from both the left and the right, we will notice a pattern in the results of our function, \(\frac{\sqrt{x+1}-1}{x}\).
  • Choose values like \(x = -0.1, -0.01, 0.01, 0.1\) and so on.
  • Substitute into the function and calculate the output.
  • Observe how these outputs behave as \(x\) approaches zero.
You will notice the function values appear to approach a specific value. This is the limit of the function as \(x\) approaches zero, hinting that our calculated estimate is getting closer to the actual limit value. This hands-on approach reinforces intuition and understanding of how limits function.
Rationalizing the Numerator
Rationalizing deals with simplifying a fraction to remove radicals from the numerator. For our limit problem \(\lim_{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x}\), the radical is in the numerator. This can make direct limit calculation difficult.To rationalize:
  • Multiply the numerator \(\sqrt{x+1}-1\) by its conjugate \(\sqrt{x+1}+1\).
  • Also multiply the denominator by this conjugate to keep the fraction unchanged.
This will convert the radical portion into a difference of squares, simplifying to \(\frac{(\sqrt{x+1})^2 - 1^2}{x(\sqrt{x+1}+1)}\), which further simplifies to \(\frac{x}{x(\sqrt{x+1}+1)}\).Once simplified, potential difficulties with radicals are removed, making evaluation easier. Rationalizing is a powerful tool to simplify expressions, paving the way for simpler limit calculations.
Graphing Functions
Graphing provides a visual comprehension of a function's behavior as it approaches a certain point, such as the limit \(\lim_{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x}\). By using graphing tools:
  • Firstly, plot the function over a range that surrounds the point of interest, \(x = 0\).
  • Examine how the graph behaves as \(x\) nears the specific value.
In this case, observe as \(x\) approaches 0 from both the left and right side. The curve should converge to a horizontal line that represents the limit value. Graphing is not just about confirming numerical results but also about gaining insights into how functions transition and transform. It visually reinforces our understanding of limit behavior.
Evaluating Limits
Evaluating limits involves finding the value that a function approaches as the input gets arbitrarily close to a given point. The function \(\frac{1}{\sqrt{x+1}+1}\) resulted from simplifying the original problem and can be evaluated directly at \(x = 0\) without any undefined expressions.Through substitution:
  • Insert \(x = 0\) into the simplified function.
  • Calculate the expression \(\frac{1}{\sqrt{0+1}+1} = \frac{1}{2}\).
This shows that the limit of the original expression as \(x\) approaches 0 is \(\frac{1}{2}\).Evaluating limits with the correct techniques and simplified forms ensures precision. By breaking down and systematically approaching each problem step, understanding is deepened, which is essential for more complex calculus concepts.

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Most popular questions from this chapter

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