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Question 1

For the following function, $f(x)$, evaluate the following limits:

1. $\displaystyle\mathop{{\rm lim}}_{x\to {-1}^-}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to {-1}^+}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to -1}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 0^-}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 0^+}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 0}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 1^-}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 1^+}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 1}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 2^-}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 2^+}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 2}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 3^-}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 3^+}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 3}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 5^-}f\left(x\right)$

1. $\displaystyle\mathop{{\rm lim}}_{x\to 5^+}f\left(x\right)$

Studyforge Question ID: 19085

Question 2

Consider the piecewise function
$f(x)=\begin{cases}\dfrac{x^4-1}{x-1} & x\neq 1\\3 & x=1\\\end{cases}$

1. a) Graph the function on the domain [-3, 3].

1. b) Find the $\displaystyle\mathop{{\rm lim}}_{x\to 1^-}f\left(x\right)$ .

1. c) Find the $\displaystyle\mathop{{\rm lim}}_{x\to 1^+}f\left(x\right)$ .

1. d) Find the $\displaystyle\mathop{{\rm lim}}_{x\to 1}f\left(x\right)$.

1. e) Find $f(1)$.

Studyforge Question ID: 151

Question 3

Sketch a possible graph of a function that has domain $-3\le x\le 5$, has zeros (intersects the x-axis) at -2, and 1, and the $\displaystyle\lim_{x\to -2}f(x)=5$, and $\displaystyle\lim_{x\to 1}f(x)=DNE$

Possible answer:

Studyforge Question ID: 153

Question 4

Sketch a possible graph of a function that has domain  $x \le 0$, but not $x=-2$,  goes through the points (-4,2) and (0,2), and has $\displaystyle\mathop{{\rm lim}}_{x\to -2}f\left(x\right)=\ \infty$.

Possible answer:

Studyforge Question ID: 154

Question 5

Sketch a possible graph of a function that has zeros of -2, and 1, and $\displaystyle\lim_{x\to -2^-}f(x)=-\infty$ , but $\displaystyle\lim_{x\to -2^+}f(x)=\infty$, and $\displaystyle\lim_{x\to 1}f(x)=3$

Possible answer:

Studyforge Question ID: 155

Question 6

Estimate the value of $\mathop{{\rm lim}}_{x\to 0}(1+x)^{\frac{1}{x}}$ by trying values very close to 0 (from both the right and the left, obviously, since it's not a one-sided limit). Once you've made a reasonable estimate, check it by actually graphing the function near that point.

Note: The limit value converges to "e"

Studyforge Question ID: 156

Question 7

Estimate the value of $\mathop{{\rm lim}}_{x\to 2^-}\frac{3x+6}{3x^3-24}$ by trying values very close to 2 (from the left only, of course). Once you've made a reasonable estimate, check it by actually graphing the function near that point.

Estimation:

Graph:

Studyforge Question ID: 157

Question 8

For the function $f(x)={3x}^2$, approximate the slope of the tangent line at (1,3) by doing the following:

1. Make a graph of f(x) and plot a point at $x=a$, somewhere nearby the point (1,3).

1. Write the slope of the secant line $m_{sec}$ that joins the point (1,3) and the point (a, f(a)) in terms of a, and simplify.

1. Take the $\mathop{{\rm lim}}_{a\to 1}\ m_{sec}$ to determine the actual slope of the tangent line at (1,3).

Studyforge Question ID: 158

Question 9

For the function $f\left(x\right)=-2x^2$ , approximate the slope of the tangent line at (1,- 2) using the secant line approximation method (see question ID 158 for the individual steps).

Studyforge Question ID: 159

Question 10

Evaluate the following limit:

$\displaystyle\mathop{{\rm lim}}_{x\to 5}\ \ 3x$

Studyforge Question ID: 160

Question 11

Evaluate this limit:

$\displaystyle\mathop{{\rm lim}}_{x\to \pi }\ \ 4x^2$

Studyforge Question ID: 161

Question 12

Evaluate the following limit:

$\displaystyle\mathop{{\rm lim}}_{x\to 0}\frac{{\rm 8}}{x}$

Studyforge Question ID: 162

Question 13

Evaluate this limit:

$\displaystyle\lim_{x\to 0}\frac{\cos{(x)}}{x}$

Studyforge Question ID: 163

Question 14

Evaluate this limit:

$\displaystyle\lim_{x\to 0^+}\frac{5}{x^2}$

Studyforge Question ID: 164

Question 15

Evaluate the following limit:

$\displaystyle\lim_{x\to 0^-}\frac{10}{x^3}$

Studyforge Question ID: 165

Question 16

AP Prep:

Which of the following limits exist?

1. $\lim_{x\to -5^{-}}{f(x)}$

1. $\lim_{x\to -5^{+}}{f(x)}$

1. $\lim_{x\to -5}{f(x)}$

1. $\lim_{x\to -3^{-}}{f(x)}$

1. $\lim_{x\to -3^{+}}{f(x)}$

1. $\lim_{x\to -3}{f(x)}$

1. $\lim_{x\to 0^{-}}{f(x)}$

1. $\lim_{x\to 0^{+}}{f(x)}$

1. $\lim_{x\to 0}{f(x)}$

1. $\lim_{x\to 3^{-}}{f(x)}$

1. $\lim_{x\to 3^{+}}{f(x)}$

1. $\lim_{x\to 3}{f(x)}$

1. $\lim_{x\to 5^{-}}{f(x)}$

1. $\lim_{x\to 5^{+}}{f(x)}$

1. $\lim_{x\to 5}{f(x)}$

Studyforge Question ID: 3207

Question 17

AP Prep:

Consider the function:

$f(x)=\begin{cases}x^2+2 & x<3\\x & x\ge 3\end{cases}$

Which of the following are true?

1. $f(3)=11$

1. $\lim_{x\to 3}f(x)=3$

1. $\lim_{x\to 3^{-}}f(x)$ exists.

1. $\lim_{x\to 3}f(x)$ exists.

1. $\lim_{x\to 3^{+}}f(x)=3$

1. $\lim_{x\to 3^{+}}f(x)=f(3)$

Studyforge Question ID: 9149

Question 18

[Try This!]:

What is the formal definition of a limit of a real function? This is also known as the epsilon-delta definition of a limit (hint: wikipedia!). Once you've got the definition, try to figure out what it means, and see if you can use it to show that $\lim_{x\to 2}{2x+2}=6$.

Studyforge Question ID: 3208

Question 19

[Try This!]:

What is the formal definition of a limit of a real function? This is also known as the epsilon-delta definition of a limit (hint: wikipedia!). Once you've got the definition, try to figure out what it means, and see if you can use it to show that $\lim_{x\to 1}{3x-2}=1$.

Studyforge Question ID: 9130

Question 20

[Try This!]:

$\lim_{x\to c}f(x)=L$ is equivalent to:

(a) For all $\delta$, there is an $\epsilon$ such that if $|x-c|<\delta$, then $|f(x)-L|<\epsilon$.

(b) For all $\epsilon$, there is a $\delta$ such that if $|x-c|>\delta$, then $|f(x)-L|<\epsilon$.

(c) For all $\epsilon$, there is a $\delta$ such that if $|x-c|<\delta$, then $|f(x)-L|<\epsilon$.

(d) For all $\delta$, there is an $\epsilon$ such that if $|f(x)-L|<\epsilon$, then $|x-c|<\delta$.

Studyforge Question ID: 9148
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