1. Limits and Continuity

Given the function $f\left(x\right)=x$, find the largest value of $\delta$ such that if $|x-4|<\delta |$, then $|x-2|<0.4|$.

Theory and Examples

If limx→4 (f(x) − 5) / (x − 2) = 1, find limx→4 f(x).

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))

Find the limit.

$\(\lim\)_{x\(\rarr\)2}\(\sqrt{x^2+5}\)$

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^x)

Find the limit.

$\(\lim\)_{x\(\rarr\)0}\(\frac{\sqrt{x+1}\)-1}{x}$

Explain why lim x→3^+ √ x−3 / 2−x does not exist.

Find the limit.

$\(\lim\)_{x\(\rarr\)0}\(\frac{3x^2+7x}{x}\)$

Determine the following limits.

lim x→∞ (3 tan^-1 x + 2)

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

a. x² + 4x

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→1 x^2 − 1 / x − 1

Evaluate each limit and justify your answer. 

lim x→5 ln 6(√x^2−16−3) / 5x−25

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\underset{x\to \infty }{\lim }x\cos \left(x\right)$

Determine the following limits.

lim θ→π/2 sin^2 θ − 5 sin θ + 4 / sin^2 θ − 1

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

c. lim x→0^− f(x)