Given $f\left(x\right)=x^2$, what is the average rate of change of $f\left(x\right)$ over the interval $[4,6]$?

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(x² + 3x) − √(x² − 2x))

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(x² + x) − √(x² − x))

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 0 (−1 / x²) = −∞

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 3 (−2 / (x − 3)²) = −∞

Given the function $f\left(x\right)=x^2$, what is the average rate of change of $f\left(x\right)$ between $x=1$ and $x=2$?

For the function $f\left(x\right)=2x^2+3x$, what is the average rate of change from $x=0$ to $x=5$?

For the function $f\left(x\right)=2x$, what is the limit of the average rate of change of $f\left(x\right)$ as $x$ increases from $2$ to $4$?

If a function $f$ is continuous on $(-\infty ,\infty )$, which of the following statements is always true?