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Chapter 8: Problem 16

Compute the limits. $$ \lim _{x \rightarrow 0} \frac{x^{2}}{\sqrt{2 x+1}-1} $$

Short Answer

Expert verified
The solution to the limit is \(0\).

Step by step solution

01

- Rationalization

Transform the expression through rationalization, i.e., multiply and divide by the conjugate of the denominator. We get:\[\lim _{x \rightarrow 0} \frac{x^{2}(\sqrt{2 x+1}+1)}{(\sqrt{2 x+1}-1) (\sqrt{2 x+1}+1)}\]This simplifies to:\[\lim _{x \rightarrow 0} \frac{x^{2}(\sqrt{2 x+1}+1)}{{2 x+1}-1}\]
02

- Simplification

Now, simplify the above expression. We get:\[\lim _{x \rightarrow 0} \frac{x^{2}(\sqrt{2 x+1}+1)}{2 x}\]Then:\[\lim _{x \rightarrow 0} x(\sqrt{2x +1}+1)\]
03

- Substitution

Now, substitute \(x = 0\) into the simplified equation. We get:\[0 * (\sqrt{2*0 +1}+1) = 0\]

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Key Concepts

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

Rationalization
When dealing with limits in calculus, particularly those that involve square roots or irrational expressions, rationalization is a helpful technique. The idea is to eliminate radicals from the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator.
For example, given the limit \( \lim _{x \rightarrow 0} \frac{x^{2}}{\sqrt{2 x+1}-1} \), the denominator is \( \sqrt{2 x+1}-1 \).
  • The conjugate of \( \sqrt{2 x+1}-1 \) is \( \sqrt{2 x+1}+1 \).
  • By multiplying and dividing by this conjugate, we eliminate the radical in the denominator.
The expression becomes \( \lim _{x \rightarrow 0} \frac{x^{2}(\sqrt{2 x+1}+1)}{(\sqrt{2 x+1}-1)(\sqrt{2 x+1}+1)} \).
This now leads to the difference of squares in the denominator, simplifying it considerably.
Simplification process
In the simplification process, the goal is to reduce the complexity of the rationalized expression. After rationalization, the denominator becomes \( 2x \) because:
  • The denominator becomes \( (2x + 1) - 1 \), which simplifies to \( 2x \).
  • This simplification is crucial as it removes the initially problematic form that caused issues with direct substitution.
Thus, the expression now is \( \lim _{x \rightarrow 0} \frac{x^{2}(\sqrt{2x+1}+1)}{2x} \),
which can further be simplified to \( \lim _{x \rightarrow 0} x(\sqrt{2x+1}+1) \).
At each step, maintaining an equal expression ensures that no incorrect transformations are applied.
Substitution method
Finally, once the expression has been fully simplified, it's time to apply the substitution method to find the limit.
This step is straightforward:
  • Substitute the value, \( x = 0 \), into the simplified expression.
  • The expression is now \( 0 * (\sqrt{2*0 +1}+1) \).
When evaluated, \( \sqrt{1}+1 \) equals 2. However, the multiplication by 0 results in the whole expression becoming 0.
Thus, the limit is \( 0 \). This confirms that even if an expression seems complex initially, through careful rationalization and simplification, we can easily find its limit using substitution.

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