Limits

Explain in your own words the types of functions whose limits we can calculate with the limit rules in this section.

Explain why we can’t calculate every limit $\underset{x\to c}{\lim }f\left(x\right)$ just by evaluating f(x) at $x = c$. Support your argument with the graph of a function f for which $\underset{x\to c}{\lim }f\left(x\right)=f\left(c\right)$.

Find functions f and g and a real number c such that $\underset{x\to c}{\lim }f\left(x\right)+\underset{x\to c}{\lim }g\left(x\right)\ne \underset{x\to c}{\lim }\left(f\right(x)+g(x\left)\right)$. Does this example contradict the sum rule for limits? Why or why not?

Find functions f and g and a real number c such that $\left(\underset{x\to c}{\lim }f\left(x\right)\right)\left(\underset{x\to c}{\lim }g\left(x\right)\right)\ne \underset{x\to c}{\lim }\left(f\right(x\left)g\right(x\left)\right)$. Does this example contradict the product rule for limits? Why or why not?

Write the constant multiple rule for limits in terms of delta–epsilon statements.

Write the difference rule for limits in terms of delta–epsilon statements.

Write the product rule for limits in terms of delta–epsilon statements. 

Explain how the algebraic function $f\left(x\right)={\left(\sqrt{x}+1\right)}^3$ is a combination of identity, constant, and power functions. Why does this mean that we can calculate limits of this function at domain points by evaluation? 

Explain how the algebraic function $f\left(x\right)=\frac{\left(x^2+1\right)\left(4-3x\right)}{3x^2}$ is a combination of identity, constant, and power functions. Why does this mean that we can calculate limits of this function at domain points by evaluation? 

Suppose f and g are functions such that $\underset{x\to 3}{\lim }f\left(x\right)=5,\underset{x\to 4}{\lim }f\left(x\right)=2$ and $\underset{x\to 3}{\lim }g\left(x\right)=4$

Given this information, calcuate the limits that follow, if possible. If it is not possible with the given information, explain why. 

$\underset{x\to 3}{\lim }\left(2f\left(x\right)-3g\left(x\right)\right)$

Suppose f and g are functions such that $\underset{x\to 3}{\lim }f\left(x\right)=5,\underset{x\to 4}{\lim }f\left(x\right)=2$ and $\underset{x\to 3}{\lim }g\left(x\right)=4$

Given this information, calcuate the limits that follow, if possible. If it is not possible with the given information, explain why. 

$\underset{x\to 7}{\lim }f\left(x\right)$

Suppose f and g are functions such that $\underset{x\to 3}{\lim }f\left(x\right)=5,\underset{x\to 4}{\lim }f\left(x\right)=2$ and $\underset{x\to 3}{\lim }g\left(x\right)=4$

Given this information, calcuate the limits that follow, if possible. If it is not possible with the given information, explain why.

$\underset{x\to 4}{\lim }f\left(x\right)g\left(x\right)$

Suppose f and g are functions such that $\underset{x\to 3}{\lim }f\left(x\right)=5,\underset{x\to 4}{\lim }f\left(x\right)=2$ and $\underset{x\to 3}{\lim }g\left(x\right)=4$

Given this information, calcuate the limits that follow, if possible. If it is not possible with the given information, explain why.

$\underset{x\to 3}{\lim }\frac{f\left(x\right)-3}{g\left(x\right)}$

Suppose f and g are functions such that $\underset{x\to 3}{lim} f\left(x\right)=5$, $\underset{x\to 4}{lim} f\left(x\right)=2$, and $\underset{x\to 3}{lim} g\left(x\right)=4$. Given this information, calculate the limits that follow, if possible. If it is not possible with the given information, explain why. 

Graph the functions $f\left(x\right)=x+1$ and $g\left(x\right)=\frac{x^2-1}{x-1}$, and show that they are equal everywhere except at one point. Then show that f(x) and g(x) have different values, but the same limit, at this point. 

Graph the functions $f\left(x\right)=2-x$ and $g\left(x\right)=\frac{4-x^2}{x+2}$, and show that they are equal everywhere except at one point. Then show that f(x) and g(x) have different values, but the same limit, at this point. 

In the Squeeze Theorem for limits, we require that l(x) ≤ f(x) ≤ u(x) for all x sufficiently close to c, but we do not require this inequality to hold at the point x = c. Why not? 

Use a geometric argument and the Squeeze Theorem for limits to argue that $\underset{\theta \to 0^{-}}{lim} \sin \theta =0$

for sufficiently small negative angles θ.

Use a geometric argument and the Squeeze Theorem for limits to argue that $\underset{\theta \to 0^{-}}{lim} \cos \theta =1$

for sufficiently small negative angles θ. 

In this exercise you will use a calculator to investigate the number e.

1. Make a table of values that describes the behavior of the quantity $(1+h{)}^{1/h}$ as $h\to 0$.
2. Make a table of values that describes the behavior of the quantity $\frac{e^h-1}{h}$ as $h\to 0$.
3. What do your tables of values have to do with Definition 1.25 and Theorem 1.26?

Calculate the limits in Exercises 25–28, using only the continuity of linear and power functions and the limit rules. Cite each limit rule that you apply. 

$\underset{x\to 1}{lim} 15(3-2x)$

Calculate the limits in Exercises 25–28, using only the continuity of linear and power functions and the limit rules. Cite each limit rule that you apply. 

$\underset{x\to -1}{lim} \frac{x-1}{(x+4)(x+2)}$

Calculate the limits using only the continuity of linear and power functions and the limit rules. Cite each limit rule that you apply.

$\underset{x\to 3}{\lim }\left(3x+x^2\left(2x+1\right)\right)$

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 0}{\lim } \left(x^2-1\right)$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 2}{\lim } (x - 1)(x + 1)(x + 5)$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to -1}{\lim } \left(x^2-\frac{3x}{x+2}\right)$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 1.7}{\lim } \left(3.1x^2-4x+0.8\right)$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 2}{\lim } \frac{4+2x}{x+2}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to -2}{\lim }\frac{4+2x}{x+2}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 1}{\lim } \frac{x^2-1}{x-1}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 0}{\lim } \frac{x^2-1}{x-1}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to -3}{\lim } \frac{x+3}{3x^2+8x-3}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to -3}{\lim } \frac{3x^2+8x-3}{x+3}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\lim }\frac{4x-2}{6x^2+x-2}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\lim }\frac{6x^2+x-2}{3x+2}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 1^{+}}{\lim }\frac{x-1}{\sqrt{x-1}}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 1^{+}}{\lim }\frac{1-\sqrt{x}}{1-x}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 1}{\lim }\frac{x+x^2-2x^3}{x-x^2}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 0}{\lim }\frac{x+x^2-2x^3}{x-x^2}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{h\to 0}{\lim }\frac{{\left(1+h\right)}^2-1}{h}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{h\to 0}{\lim }\frac{{\left(-1+h\right)}^2-1}{h}$.

Calculate each of the limits in Exercises $29-70$.

$\underset{x\to 0}{\lim }\frac{2^x-3^x}{4^x}$.

$\underset{x\to +2}{\lim }\sqrt{2^x-4}$

$\underset{x\to 4}{\lim }\left(3e^{1.7x}+1\right)$

$\underset{x\to 0+}{\lim }\left(ln\left(1+\sqrt{x}\right)\right)$

$\underset{x\to 0}{\lim }\left(\frac{e^x-1}{e^{2x}+2e^x-3}\right)$

$\underset{x\to 1}{\lim }\left(\frac{e^x-1}{e^{2x}+2e^x-3}\right)$

$\underset{x\to 0}{\lim }\left(\frac{2x}{e^x-1}\right)$

$\underset{x\to \mathrm{\pi }}{\lim }\frac{1}{\cos ec(\mathrm{\pi }-\mathrm{x})}$

$\underset{x\to 1}{\lim }\frac{1-\cos (x-1)}{x}$