Chapter 13: Limits: A preview of Calculus

When we write $\underset{x\to a}{\lim }f\left(x\right)=L$

then, roughly speaking, the values of $f\left(x\right)$

get closer and closer to the number ____ as the value of $x$

get closer and closer to ____.

To determine $\underset{x\to 5}{\lim }\frac{x-5}{x-5}$

, we try values for $x$

closer and closer to ____ and find that the limit is ____.

Suppose the following limit exist:

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

$\begin{array}{l}Then\underset{x\to a}{\lim }\left[f\right(x)+g(x\left)\right]=\_\_\_\_\_\_+\_\_\_\_\_\_,\text{and}\\ \underset{x\to a}{\lim }\left[f\right(x\left)g\right(x\left)\right]=\_\_\_\_\_\_+\_\_\_\_\_\_\end{array}$

These formulas can be stated verbally as follow: The limit of a sum is the _____ of the limits, and the limit of a product is the ______ of the limits.

The derivative of a function fat a number ais

$f\text{'}\left(a\right)=\underset{h\to 0}{\lim }\frac{\begin{array}{c}\_\_\_\end{array}-\_\_\_}{\_\_\_}$

if the limit exists. The derivative$f\text{'}\left(a\right)$is theof the tangent line to the curve$y=f\left(x\right)$at the point$(\_\_,\_\_)$.

Let f be a function defined on some interval $(a,\infty )$. Then

$\underset{x\to \infty }{\lim }f\left(x\right)=L$

Then means that the values of $f\left(x\right)$can be made arbitrarily close to _____ by taking _____ sufficiently large. In this case the line $y=L$is called a ___ _____ of the function $y=f\left(x\right)$is called a. For example, $\underset{x\to \infty }{\lim }\frac{1}{x}=$_____, and the line y=_____ is a horizontal asymptote.

When we write $\underset{x\to a}{\lim }f\left(x\right)=L$then, roughly speaking, the values of$f\left(x\right)$get closer and closer to the number ____ as the value of x get closer and closer to ____.

To determine $\underset{x\to 5}{\lim }\frac{x-5}{x-5}$, we try values for x closer and closer to ____ and find that the limit is ____.

Complete the table of values (to five decimal places), and use the table to estimate the value of the limit.

$\underset{x\to 0^{+}}{\lim }x\ln x$

Evaluate the limit and justify each step by indicating the appropriate limit law(s).

$\underset{x\to 0}{\lim }(3x^3-2x^2+5)$

Find the slope of the tangent line to the graph of fat the given point.$f\left(x\right)=\frac{6}{x+1},at(2,2)$.

find the limit.

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

Evaluate the limit and justify each step by indicating the appropriate limit law(s).

$\underset{x\to -1}{\lim }\frac{x-2}{x^2+4x-3}$