Finding Limits of Differences When x → ±∞


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


lim x → ∞ (√(x² + 25) − √(x² − 1))

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = (x² − 4) / (x − 1)

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = (x² − 1) / x

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 (1 + x + sin x) / (3 cosx)

Additional Graphing Exercises

[Technology Exercise] Graph the curves in Exercises 109–112. Explain the relationship between the curve’s formula and what you see.

y = −1 / √(4 − x²)

Given the function $f\left(x\right)=2x^2-3x+1$, which of the following is the average rate of change of $f\left(x\right)$ over the interval from $x=-6$ to $x=-3$?

How many tangent lines to the curve $y=\frac{x}{x+1}$ pass through the point $(1,\frac{1}{2})$?

At what point do the curves $r_1\left(t\right)=(t,2-t,35+t^2)$ and $r_2\left(s\right)=(7-s,s-5,s^2)$ intersect?

Given that $d\left(t\right)$ is a known function and $d$ is a specific value, which of the following procedures will find the time $t$ at which $d\left(t\right)=d$?