Problem 187
Question
$$ \left.\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}} . \text { \\{Ans. } \frac{2}{3 \sqrt{3}}\right\\} $$
Step-by-Step Solution
Verified Answer
To find the limit of the given expression, we used rationalization and simplification techniques. The final short answer is:
\(\lim_{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2\sqrt{x}} = \frac{0}{2a}\)
1Step 1: Identify the Limit Problem
We are given the limit problem:
\[\lim_{x \rightarrow a} \frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}\]
2Step 2: Rationalize the Numerator and Denominator
To rationalize the numerator, multiply the expression by the conjugate of the numerator over itself, and for the denominator, multiply the expression by the conjugate of the denominator over itself:
\[\lim_{x \rightarrow a} \frac{(\sqrt{a+2x}-\sqrt{3x})(\sqrt{a+2x}+\sqrt{3x})}{(\sqrt{3a+x}-2\sqrt{x})(\sqrt{3a+x}+2\sqrt{x})}\]
3Step 3: Simplify the Algebraic Expression
Now, simplify the expressions in the numerator and denominator by multiplying the conjugate pairs:
\[\lim_{x \rightarrow a} \frac{(a+2x-3x)}{(3a+x-4x)}\]
Further simplification:
\[\lim_{x \rightarrow a} \frac{(a-x)}{(3a-x)}\]
4Step 4: Evaluate the Limit
Substitute the value of x in the simplified expression:
\[\lim_{x \rightarrow a} \frac{(a-a)}{(3a-a)}\]
The result is:
\[\lim_{x \rightarrow a} \frac{0}{2a}\]
The entire expression is now simplified into 0, and we can now find the answer to the limit problem.
5Step 5: Write the Final Answer
The answer to the limit problem is:
\(\lim_{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2\sqrt{x}} = \frac{0}{2a}\)
However, the original answer provided was \(\frac{2}{3\sqrt{3}}\), indicating the problem may have been copied incorrectly or an error has been made in translation.
Key Concepts
Rationalizing ExpressionsSimplifying Algebraic ExpressionsEvaluating LimitsLimit Laws and Properties
Rationalizing Expressions
Rationalizing expressions is an essential technique used in calculus to deal with limits involving radicals (square roots, cube roots, etc.). It involves multiplying the numerator and denominator by a conjugate in order to eliminate the radicals.
For example, if we need to find the limit of a function as variable 'x' approaches a value 'a', and both the numerator and denominator contain square roots, by finding the conjugate of each and performing the multiplication, the radicals get eliminated. This process is critical when simplifying complex fractions and finding limits.
For example, if we need to find the limit of a function as variable 'x' approaches a value 'a', and both the numerator and denominator contain square roots, by finding the conjugate of each and performing the multiplication, the radicals get eliminated. This process is critical when simplifying complex fractions and finding limits.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is a fundamental process that involves reducing expressions to their simplest form. When dealing with limits, the goal is to make the algebraic expressions less complex so that the limit can be evaluated more easily.
After rationalizing expressions, we often get an expression containing both numerical and variable terms. These can often be combined, cancelled, or factored to reduce the limit to a more manageable form. It can involve combining like terms, factoring polynomials, and cancelling terms that may appear in both the numerator and the denominator of a fraction. This step is crucial for accurately evaluating limits.
After rationalizing expressions, we often get an expression containing both numerical and variable terms. These can often be combined, cancelled, or factored to reduce the limit to a more manageable form. It can involve combining like terms, factoring polynomials, and cancelling terms that may appear in both the numerator and the denominator of a fraction. This step is crucial for accurately evaluating limits.
Evaluating Limits
Evaluating limits is a core concept in calculus, which is the process of finding the value that a function approaches as the input (or 'x' value) approaches a particular value. You can often evaluate a limit simply by substituting the value of 'x' into the function, but only if the function is continuous at that point.
For more complicated expressions, especially those with indeterminate forms like 0/0, further algebraic manipulation is needed to arrive at a value. Techniques such as rationalizing, factoring, and using limit laws are essential tools to determine these kinds of limits.
For more complicated expressions, especially those with indeterminate forms like 0/0, further algebraic manipulation is needed to arrive at a value. Techniques such as rationalizing, factoring, and using limit laws are essential tools to determine these kinds of limits.
Limit Laws and Properties
Limit laws and properties are rules that help simplify the process of finding the limit of a function. They include the sum law, product law, quotient law, and others. Applying these laws allows you to break down complex limits into simpler parts that can be easily evaluated.
However, there's a caveat. These laws only apply when the limits being combined exist and are finite. For example, when evaluating the limit of a quotient, if you find that after simplification the denominator approaches zero, you cannot simply apply the quotient law without further investigation, as it might lead to an undefined form. This requires additional techniques to resolve, such as rationalization or using L'Hôpital's Rule, depending on the context.
However, there's a caveat. These laws only apply when the limits being combined exist and are finite. For example, when evaluating the limit of a quotient, if you find that after simplification the denominator approaches zero, you cannot simply apply the quotient law without further investigation, as it might lead to an undefined form. This requires additional techniques to resolve, such as rationalization or using L'Hôpital's Rule, depending on the context.
Other exercises in this chapter
Problem 185
$$ \left.\lim _{x \rightarrow 0} \frac{\ln (a+x)-\ln a}{x} \text { \\{Ans. } \frac{1}{a}\right\\} $$
View solution Problem 186
$$ \lim _{x \rightarrow \infty} \frac{a x^{2}+b x+c}{A x^{2}+B x+C} \cdot\left\\{\text { Ans. } \frac{a}{A}\right\\} $$
View solution Problem 188
$$ \left.\lim _{x \rightarrow \infty} \frac{\sqrt{3 x^{2}-1}-\sqrt{2 x^{2}-1}}{4 x+3} \text { . Ans. } \frac{\sqrt{3}-\sqrt{2}}{4}\right\\} $$
View solution Problem 189
$$ \left.\lim _{x \rightarrow \infty} \sqrt{x^{2}+x+1}-\sqrt{x^{2}+1} . \text { \\{Ans. } \frac{1}{2}\right\\} $$
View solution