Exercises


  1. Given that \( \displaystyle \lim_{x \rightarrow a} ~ f(x) = 4 \) and \( \displaystyle \lim_{x \rightarrow a} ~ g(x) = -2 \), find the following limits:
    1. \( \displaystyle \lim_{x \rightarrow a} ~ \left( f(x) + g(x) \right) \)
    2. \( \displaystyle \lim_{x \rightarrow a} ~ f(x)g(x) \)
    3. \( \displaystyle \lim_{x \rightarrow a} ~ 4 f(x) \)
    4. \( \displaystyle \lim_{x \rightarrow a} ~ \dfrac{f(x)}{g(x)} \)
    5. \( \displaystyle \lim_{x \rightarrow a} ~ \dfrac{\sqrt{f(x)}}{g(x)} \)
    6. \( \displaystyle \lim_{x \rightarrow a} ~ \dfrac{f(x) + 2}{2 - 2g(x)} \)
  2. Find the following limits.
    1. \( \displaystyle \lim_{x \rightarrow 5} ~ (2x + 3) \)
    2. \( \displaystyle \lim_{x \rightarrow 2} ~ (-x^2 + 3x - 2) \)
    3. \( \displaystyle \lim_{t \rightarrow -1} ~ 5(t - 2)(t - 3) \)
    4. \( \displaystyle \lim_{x \rightarrow 3} ~ \dfrac{x + 3}{x - 2} \)
    5. \( \displaystyle \lim_{x \rightarrow 0} ~ 2(2x - 1)^3 \)
    6. \( \displaystyle \lim_{x \rightarrow -3} ~ \left( 5 - x \right)^{\frac{4}{3}} \)
  3. Suppose we have that \(\displaystyle \lim_{x \rightarrow 0} ~ f(x) = 2 \) and \(\displaystyle \lim_{x \rightarrow 0} ~ g(x) = -10 \). State the limit properties that are used to accomplish steps (a), (b), and (c) of the following calculation:\[\begin{align*} \lim_{x \rightarrow 0} ~ \dfrac{3f(x) - g(x)}{(g(x) + 2)^{\frac{1}{3}}} &= \dfrac{\left( \displaystyle \lim_{x \rightarrow 0} \left( 3f(x) - g(x) \right) \right)}{\displaystyle \lim_{x \rightarrow 0} \left( g(x) + 2 \right)^{\frac{1}{3}} } \tag{a} \\ &= \dfrac{\displaystyle \lim_{x \rightarrow 0} 3f(x) - \displaystyle \lim_{x \rightarrow 0} g(x)}{\displaystyle \left( \lim_{x \rightarrow 0} \left( g(x) + 2 \right) \right)^{\frac{1}{3}} } \tag{b} \\ &= \dfrac{\displaystyle 3\lim_{x \rightarrow 0} f(x) - \lim_{x \rightarrow 0} g(x) }{\displaystyle \left( \lim_{x \rightarrow 0} g(x) + \lim_{x \rightarrow 0} 2 \right)^{\frac{1}{3}} } \tag{c} \\ &= \dfrac{ 3(2) - (-10) }{\left( -10 + 2 \right)^{\frac{1}{3}} } \\ &= \dfrac{16}{-2} \\ &= -8 \end{align*}\]
  4. If \( \displaystyle \lim_{x \rightarrow 1} ~ f(x) = -2 \) and \( \displaystyle \lim_{x \rightarrow 1} ~ g(x) = 3 \), then what is the value of \( \displaystyle \lim_{x \rightarrow 1} \dfrac{ [f(x)]^3 + [g(x)]^2 }{5 - 2g(x)} \)?
  5. If \( \displaystyle \lim_{x \rightarrow a} ~ f(x) = 4 \) and \( \displaystyle \lim_{x \rightarrow a} ~ g(x) = -1 \), then what is the value of \( \displaystyle \lim_{x \rightarrow a} \sqrt{\dfrac{ 2\sqrt{5 + g(x)} }{3f(x) + 2g(x) + 6}} \)?
  6. Line f through (a, f(a)), (b, 6); line g through (a, f(a)), (b, 2) Let \( a \lt b \) be real numbers. Consider two linear functions as shown in the graph. Evaluate \( \displaystyle \lim_{x \rightarrow a} ~ \dfrac{f(x)}{g(x)} \).
     
  7. Give an example of functions \( f(x) \) and \( g(x) \) such that \( \displaystyle \lim_{x \rightarrow 0} \left( f(x) + g(x) \right) \) exists but \( \displaystyle \lim_{x \rightarrow 0} ~ f(x) \) and \( \displaystyle \lim_{x \rightarrow 0} ~ g(x) \) do not exist.
  8. Give an example of a function such that \( \displaystyle \lim_{x \rightarrow 0} ~ [f(x)]^2 \) exists but \( \displaystyle \lim_{x \rightarrow 0} ~ f(x) \) does not exist.