Review and Practice of Limits

Review and Practice of Limits

1) You get the answer and you're done

2) You get 0/0 and need to do more work:

• factor/cancel

• get a com denom

• rationalize

• Special Trig limits

• L'Hopital's Rule

3) You get n/0:

(a nonzero number divided by 0)

This indicates a vertical asymptote on the graph of f(x) at x = c

So limit = ∞ or -∞

You would need to plug in an x-value close to "c" to determine the correct sign.

4) If nothing's working try graphing or plugging in x-values close to "c

Determine which parts of the function f(x) will grow the fastest (or be the largest) as x becomes infinitely large. If your function is a fraction, you can consider the following cases:

CASE ONE: Small/Large

Limit = 0

This is a hor. asymp.

CASE TWO: Large/Small

Limit = ∞

Or Limit = −∞

You need to plug in a large value to determine the correct sign

CASE THREE: Same/Same

Limit = a/b

Where a & b are the leading coefficients of the numerator and denominator respectively

This is a hor. asymp.

You can also use L'Hopital's Rule for any scenario where you have ∞/∞

​Part 3: The Formal Definition (or Limit Definition) of a Derivative

There are 2 main limit definitions of the derivative:

Limits that indicate asymptotes

​Limits as x →∞ :

Multiple Choice

Evaluate: $$\lim_{x\rightarrow2}\ \frac{x^2+x-6}{x^2-4}=$$

• $$-\frac{1}{4}$$

• 0

• 1

• $$\frac{5}{4}$$

• dne

Multiple Choice

Evaluate: $$\lim_{x\rightarrow-3}\ \frac{x^2-9}{x^2-2x-15}$$

• 0

• $$\frac{3}{5}$$

• $$\frac{3}{4}$$

• 1

• dne

Multiple Choice

For which of the following does $$\lim_{x\rightarrow\infty}\ f\left(x\right)\ =\ 0$$ ?

I. $$f\left(x\right)=\frac{\ln x}{x^{99}}$$

II. $$f\left(x\right)=\frac{e^x}{\ln x}$$

III. $$f\left(x\right)=\frac{x^{99}}{e^x}$$

• I. only

• II. only

• III. only

• I. and II. only

• I. and III. only

Multiple Choice

Evaluate: $$\lim_{x\rightarrow\infty}\ \frac{x^3}{e^{3x}}=$$

• 0

• $$\frac{2}{9}$$

• $$\frac{2}{3}$$

• 1

• infinite

Multiple Choice

Evaluate: $$\lim_{x\rightarrow-\infty}\ \frac{x^3}{e^{3x}}=$$

• 0

• $$\infty$$

• $$-\infty$$

• 1

Multiple Choice

$$\lim_{x\rightarrow\infty}\ \frac{\sqrt[]{9x^4+1}}{4x^2+3}$$ =

• $$\frac{1}{3}$$

• $$\frac{3}{4}$$

• $$\frac{3}{2}$$

• $$\frac{9}{4}$$

• $$\infty$$

Multiple Choice

For which of the following pairs of functions $$f$$ and $$g$$ is $$\lim_{x\rightarrow\infty}\ \frac{f\left(x\right)}{g\left(x\right)}$$ infinite?

• $$f\left(x\right)=x^2+2x$$ and $$g\left(x\right)=x^2+\ln x$$

• $$f\left(x\right)=3x^3$$

  and

  $$g\left(x\right)=x^4$$

• $$f\left(x\right)=3^x$$

  and

  $$g\left(x\right)=x^3$$

• $$f\left(x\right)=3e^x+x^3$$ and

  $$g\left(x\right)=2e^x+x^2$$

Multiple Choice

Evaluate: $$\lim_{x\rightarrow3^-}\ \frac{\left|x-3\right|}{x-3}\ =$$

• -3

• -1

• 1

• 3

• dne

Multiple Choice

Evaluate: $$\lim_{x\rightarrow2}\ \frac{\ln\left(x+3\right)-\ln\left(5\right)}{x-2}$$

• 0

• $$\frac{1}{5}$$

• $$\frac{1}{2}$$

• 1

• dne

Multiple Choice

Evaluate: $$\lim_{x\rightarrow\frac{\pi}{2}}\ \frac{\cos\left(x\right)-\cos\left(\frac{\pi}{2}\right)}{x-\frac{\pi}{2}}$$ =

• 0

• dne

• 1

• -1

Multiple Choice

$$\lim_{h\rightarrow0}\ \frac{\sin\left(\frac{\pi}{3}+h\right)-\sin\left(\frac{\pi}{3}\right)}{h}$$ =

• 0

• $$\frac{1}{2}$$

• 1

• $$\frac{\sqrt[]{3}}{2}$$

• dne

Multiple Choice

Evaluate $$\lim_{h\rightarrow0}\ \frac{\arcsin\left(a+h\right)-\arcsin\left(a\right)}{h}=2$$ , which of the following could be the value of a?

• $$\frac{\sqrt[]{2}}{2}$$

• $$\frac{\sqrt[]{3}}{2}$$

• $$\sqrt[]{3}$$

• $$\frac{1}{2}$$

• 2

Multiple Choice

The vertical line x = 2 is an asymptote for the graph of the function $$f$$ . Which of the following statements must be false?

• $$\lim_{x\rightarrow2}\ f\left(x\right)=0$$

• $$\lim_{x\rightarrow2^+}\ f\left(x\right)=-\infty$$

• $$\lim_{x\rightarrow2^-}\ f\left(x\right)\ =\ \infty$$

• $$\lim_{x\rightarrow\infty}\ f\left(x\right)=2$$

• $$\lim_{x\rightarrow\infty}\ f\left(x\right)=\infty$$

Multiple Choice

Let $$f$$ be the function given by $$f\left(x\right)=\frac{\left(x-4\right)\left(2x-3\right)}{\left(x-1\right)^2}$$ . If the line $$y=b$$ is a horizontal asymptote to the graph of $$f$$ , then $$b\ =$$

• 0

• 1

• 2

• 3

• 4

Multiple Choice

Let $$f$$ be the function defined by $$f\left(x\right)=\frac{\left(3x+8\right)\left(5-4x\right)}{\left(2x+1\right)^2}$$ . Which of the following is a horizontal asymptote to the graph of $$f$$ ?

• $$y=-6$$

• $$y=-3$$

• $$y=-\frac{1}{2}$$

• $$y=0$$

• $$y=\frac{3}{2}$$

Multiple Choice

The line $$y=5$$ is a horizontal asymptote to the graph of which of the following functions?

• $$y=\frac{\sin\left(5x\right)}{x}$$

• $$y=5x$$

• $$y=\frac{1}{x-5}$$

• $$y=\frac{5x}{1-x}$$

• $$y=\frac{20x^2-x}{1+4x^2}$$

Multiple Choice

The graph of which of the following functions has exactly one horizontal asymptote and no vertical asymptotes?

• $$y=\frac{1}{x^2+1}$$

• $$y=\frac{1}{x^3+1}$$

• $$y=\frac{1}{e^x-1}$$

• $$y=\frac{1}{e^x+1}$$

Multiple Choice

The graph of a function $$f$$ is shown above. Which of the following limits does not exist?

• $$\lim_{x\rightarrow1^-}\ f\left(x\right)$$

• $$\lim_{x\rightarrow1}\ f\left(x\right)$$

• $$\lim_{x\rightarrow3^-}\ f\left(x\right)$$

• $$\lim_{x\rightarrow3}\ f\left(x\right)$$

• $$\lim_{x\rightarrow5}\ f\left(x\right)$$

Multiple Choice

The graph of a function $$f$$ is shown in the figure above. Which of the following statements is true?

• f(a) = 2

• $$f$$ is continuous at $$x=a$$

• $$\lim_{x\rightarrow a}\ f\left(x\right)\ =\ 1$$

• $$\lim_{x\rightarrow a}\ f\left(x\right)\ =\ 2$$

• $$\lim_{x\rightarrow a}\ f\left(x\right)$$ does not exist

Multiple Choice

The figure above shows the graph of the function $$f$$ . Which of the following statements are true?

I. $$\lim_{x\rightarrow2^-}f\left(x\right)=f\left(2\right)$$

II. $$\lim_{x\rightarrow6^-}f\left(x\right)=\lim_{x\rightarrow6^+}f\left(x\right)$$

III. $$\lim_{x\rightarrow6}f\left(x\right)=f\left(6\right)$$

• II only

• III only

• I and II only

• II and III only

• I, II, and III