Salih_KARAKULLUKÇU said
Question:
Evaluate the following limit:
\lim_{h \to 0} \frac{\sqrt{h^2 + 7h + 5} - \sqrt{5}}{h}
Answer:
To evaluate the given limit, we can use the technique of multiplying and dividing by the conjugate of the numerator.
Let’s rewrite the limit:
\lim_{h \to 0} \frac{\sqrt{h^2 + 7h + 5} - \sqrt{5}}{h}
Multiply and divide by the conjugate of the numerator:
\lim_{h \to 0} \frac{\left(\sqrt{h^2 + 7h + 5} - \sqrt{5}\right) \left(\sqrt{h^2 + 7h + 5} + \sqrt{5}\right)}{h \left(\sqrt{h^2 + 7h + 5} + \sqrt{5}\right)}
The numerator becomes:
(\sqrt{h^2 + 7h + 5})^2 - (\sqrt{5})^2 = (h^2 + 7h + 5) - 5 = h^2 + 7h
Thus, the limit expression simplifies to:
\lim_{h \to 0} \frac{h^2 + 7h}{h \left(\sqrt{h^2 + 7h + 5} + \sqrt{5}\right)}
We can factor out an h from the numerator:
\lim_{h \to 0} \frac{h(h + 7)}{h \left(\sqrt{h^2 + 7h + 5} + \sqrt{5}\right)}
Cancel out the h:
\lim_{h \to 0} \frac{h + 7}{\sqrt{h^2 + 7h + 5} + \sqrt{5}}
Now, evaluate the limit as h approaches 0:
\frac{0 + 7}{\sqrt{0^2 + 0 + 5} + \sqrt{5}} = \frac{7}{2\sqrt{5}}
Simplify the fraction:
\frac{7}{2\sqrt{5}} = \frac{7\sqrt{5}}{10} = \frac{7}{\sqrt{10}}
Therefore, the correct answer is:
\boxed{\frac{7}{\sqrt{10}}}
So, option IV is the correct answer:
\boxed{IV) \frac{7}{\sqrt{10}}}