Evaluate the limit $$ \lim _{x \rightarrow \infty} \frac{\sqrt{10+7 x^{2}}}{8+10 x} $$

Question

UpStudy AI · Accepted Answer

To evaluate the limit:

$$
\lim_{x \rightarrow \infty} \frac{\sqrt{10 + 7x^2}}{8 + 10x}
$$

we analyze the behavior of both the numerator and the denominator as $$ x $$ approaches infinity.

1. **Numerator Analysis:**

$$
   \sqrt{10 + 7x^2}
   $$

As $$ x $$ becomes very large, the term $$ 7x^2 $$ dominates over 10. Therefore:

$$
   \sqrt{10 + 7x^2} \approx \sqrt{7x^2} = x\sqrt{7}
   $$

2. **Denominator Analysis:**

$$
   8 + 10x
   $$

As $$ x $$ becomes very large, the term $$ 10x $$ dominates over 8. Therefore:

$$
   8 + 10x \approx 10x
   $$

3. **Simplifying the Limit:**

Substitute the approximations back into the limit:

$$
   \lim_{x \rightarrow \infty} \frac{x\sqrt{7}}{10x}
   $$

The $$ x $$ terms cancel out:

$$
   \frac{\sqrt{7}}{10}
   $$

Therefore, the limit is:

$$
\boxed{\dfrac{\sqrt{7}}{10}}
$$
The limit is $$ \frac{\sqrt{7}}{10} $$.

Evaluate the limit \( \lim _{x \rightarrow \infty} \frac{\sqrt{10+7 x^{2}}}{8+10 x} \)