Questions: Question 8 Evaluate the following limits. If needed, (a) lim x → ∞ (sqrt(4+10 x^2))/(4+5 x) = □ (b) lim x → -∞ (sqrt(4+10 x^2))/(4+5 x) = □

Transcript text: Question 8 Evaluate the following limits. If needed, (a) $\lim _{x \rightarrow \infty} \frac{\sqrt{4+10 x^{2}}}{4+5 x}=$ $\square$ (b) $\lim _{x \rightarrow-\infty} \frac{\sqrt{4+10 x^{2}}}{4+5 x}=$ $\square$

Solution

Solution Steps

Step 1: Factor out the highest power of x

Factor out $x^2$ from the square root in the numerator and $x$ from the denominator. This simplifies the expression to $\sqrt{b + \frac{a}{x^2}}$ over $d + \frac{c}{x}$.

Step 2: Simplify the expression

As $x$ approaches $\pm\infty$, the terms $\frac{a}{x^2}$ and $\frac{c}{x}$ approach 0. Thus, the expression simplifies to $\sqrt{b}/d$.

Step 3: Evaluate the limit

For $x ightarrow \infty$, the limit is $\sqrt10/d = 0.63$.

Final Answer:

The limit as $x$ approaches +inf is 0.63.

Step 1: Factor out the highest power of x

Factor out $x^2$ from the square root in the numerator and $x$ from the denominator. This simplifies the expression to $\sqrt{b + \frac{a}{x^2}}$ over $d + \frac{c}{x}$.

Step 2: Simplify the expression

As $x$ approaches $\pm\infty$, the terms $\frac{a}{x^2}$ and $\frac{c}{x}$ approach 0. Thus, the expression simplifies to $\sqrt{b}/d$.

Step 3: Evaluate the limit

For $x ightarrow -\infty$, considering the sign of $x$ when taken out of the square root, the limit is -$\sqrt10/d = -0.63$.

Final Answer:

The limit as $x$ approaches -inf is -0.63.

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