To evaluate the limit [tex]\(\lim_{x \rightarrow 0} \frac{7x}{\sqrt{3x + 4} - 2}\)[/tex], follow these steps:
1. Identify the Indeterminate Form:
As [tex]\(x\)[/tex] approaches 0, the numerator [tex]\(7x\)[/tex] approaches 0, and the denominator [tex]\(\sqrt{3x + 4} - 2\)[/tex] approaches [tex]\(\sqrt{4} - 2 = 0\)[/tex]. Thus, we have a [tex]\(\frac{0}{0}\)[/tex] indeterminate form, so we can apply algebraic manipulation or use L'Hospital's rule to evaluate the limit.
2. Rationalize the Denominator:
To eliminate the indeterminate form, we rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(\sqrt{3x + 4} - 2\)[/tex] is [tex]\(\sqrt{3x + 4} + 2\)[/tex].
[tex]\[\lim_{x \rightarrow 0} \frac{7x}{\sqrt{3x + 4} - 2} \cdot \frac{\sqrt{3x + 4} + 2}{\sqrt{3x + 4} + 2}\][/tex]
This multiplication gives:
[tex]\[\lim_{x \rightarrow 0} \frac{7x (\sqrt{3x + 4} + 2)}{(\sqrt{3x + 4} - 2)(\sqrt{3x + 4} + 2)}\][/tex]
3. Simplify the Expression:
The denominator simplifies using the difference of squares formula [tex]\( (a - b)(a + b) = a^2 - b^2\)[/tex]:
[tex]\[\lim_{x \rightarrow 0} \frac{7x (\sqrt{3x + 4} + 2)}{(3x + 4) - 4}\][/tex]
Simplifying further:
[tex]\[\lim_{x \rightarrow 0} \frac{7x (\sqrt{3x + 4} + 2)}{3x}\][/tex]
4. Cancel the Common Terms:
We can reduce the fraction by canceling [tex]\(x\)[/tex] in the numerator and the denominator:
[tex]\[\lim_{x \rightarrow 0} \frac{7 (\sqrt{3x + 4} + 2)}{3}\][/tex]
5. Evaluate the Limit:
Now substitute [tex]\(x = 0\)[/tex] into the remaining expression:
[tex]\[\frac{7 (\sqrt{3 \cdot 0 + 4} + 2)}{3} = \frac{7 (\sqrt{4} + 2)}{3} = \frac{7 (2 + 2)}{3} = \frac{7 \cdot 4}{3} = \frac{28}{3}\][/tex]
Therefore, the limit is:
[tex]\[\lim_{x \rightarrow 0} \frac{7x}{\sqrt{3x + 4} - 2} = \frac{28}{3}\][/tex]
