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Calculus I Limits and Derivatives

Sunil Karna

Exercises 2.6 Exercise

1.

  1. If \(f(x) = x\) find \(\lim_{x \to 2} f(x)\text{.}\)
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  2. If \(f(x) = x^2 + 1\) find \(\lim_{x \to 2} f(x)\text{.}\)
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  3. If \(f(x) = \cos \frac{x}{2}\) find \(\lim_{x \to \pi} f(x)\text{.}\)
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  4. If \(f(x) = e^{-x}\) find \(\lim_{x \to \infty} f(x)\text{.}\)
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  5. If \(f(x) = x\cos x\) find \(\lim_{x \to \pi} f(x)\text{.}\)
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2.

Evaluate the Following Limits
  1. \begin{equation*} \lim_{x \to 2} (3x + 1) \end{equation*}
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  2. \begin{equation*} \lim_{x \to -1} \frac{x^2 - 1}{x + 1} \end{equation*}
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  3. \begin{equation*} \lim_{x \to 3} \frac{x^2 - 9}{x - 3} \end{equation*}
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  4. \begin{equation*} \lim_{x \to 0} \frac{2x^2 + x}{x} \end{equation*}
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  5. \begin{equation*} \lim_{x \to 4} \sqrt{x} \end{equation*}
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Answer.
  1. 7 [hint: direct substituion]
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  2. -2 [hint: indeterminate form \(\frac{0}{0}\)]
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3.

Use \(\epsilon\text{,}\) \(\delta\) definition of limit to prove the following limits:
  1. \begin{equation*} \lim_{x \to 3} (4x-1) = 11 \end{equation*}
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  2. \begin{equation*} \lim_{x \to 4} x^2 = 16 \end{equation*}
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  3. \begin{equation*} \lim_{x \to 2} \frac{x}{7}=\frac{2}{7} \end{equation*}
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  4. \begin{equation*} \lim_{x \to 2} (3x-2)=4 \end{equation*}
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  5. \begin{equation*} \lim_{x \to -1} (5x+8)=3 \end{equation*}
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4.

How close to \(2 \) do we have to take \(x\) so that \(4x-5\) is within a distance of (a) \(0.1\) and (b) \(0.01\) from \(3\text{?}\)
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5.

Evaluate:
\begin{equation*} \lim_{x \to 1} \frac{x^2-1}{x^2-x} \end{equation*}
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Answer: 2.
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6.

Evaluate:
\begin{equation*} \lim_{x\to 1} \frac{x^3-1}{x^2-1} \end{equation*}
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7.

If \(f(x) = x^2 + 2x + 1\) then find \(\lim_{x \to 2} f(x) \)
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8.

Evaluate:
  1. \begin{equation*} \lim_{x \to 4} (5x^2-2x+3) \end{equation*}
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  2. \begin{equation*} \lim_{x \to 1} \frac{x^2 - 1}{x - 1} \end{equation*}
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  3. \begin{equation*} \lim_{x \to -1} \frac{x-2}{x^2+4x-3} \end{equation*}
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  4. \begin{equation*} \lim_{x \to 1} \frac{x^2 +x-2}{x^2-3x+2} \end{equation*}
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  5. \begin{equation*} \lim_{t \to 0} \frac{\sqrt{2-t}-\sqrt{2}}{t} \end{equation*}
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  6. \begin{equation*} \lim_{x \to 1} \frac{t^3-1}{t^2-1} \end{equation*}
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  7. \begin{equation*} \lim_{x \to 2} \frac{x-\sqrt{3x-2}}{x^2-4} \end{equation*}
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  8. \begin{equation*} \lim_{x \to 4} \frac{x-4}{\sqrt{x}-2} \end{equation*}
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9.

Let
\begin{equation*} f(x) = \begin{cases} x+1; & x \lt 0 \\ 3x-5; & x \geq 0 \end{cases} \end{equation*}
then find
  1. \begin{equation*} \lim_{x \to 0^-} f(x) \end{equation*}
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  2. \(\displaystyle \lim_{x \to 0^{+}} f(x)\)
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  3. \begin{equation*} \lim_{x \to 0} f(x) \end{equation*}
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Explain if the limit exists or not.
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10.

Evaluate:
\begin{equation*} \lim_{x \to 0} \frac{\sqrt{1+x}-1}{x} \end{equation*}
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11.

Evaluate:
\begin{equation*} \lim_{x \to 5} \frac{\sqrt{4x+5}-5}{\sqrt{x-1}-2} \end{equation*}
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12.

Evaluate LHL and RHL of the given function:
  1. \begin{equation*} f(x) = \begin{cases} \frac{|x-4|}{x-4}; & x\neq 4 \\ 0; & x=4 \end{cases} \,\quad \text {at } \, x=4. \end{equation*}
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  2. \begin{equation*} f(x) = \begin{cases} 1+x^2; & \text{ if } \, 0 \leq x \leq 1 \\ 2-x; & \text{ if }\, x \gt 1 \end{cases} \,\quad \text {at } \, x=1. \end{equation*}
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13.

  1. \begin{equation*} \text{ If } f(x) = \begin{cases} 5x-4; & 0 \lt x \leq 1 \\ 4x^3-3x; & 1 \lt x \lt 2 \end{cases} \end{equation*}
    then show that \(\lim_{x \to 1} f(x)\) exist.
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  2. Prove that
    \begin{equation*} \lim_{x \to 0^+} \sqrt{x} \cos^4 x =0 \end{equation*}
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14.

Examine the continuity of the function \(f(x)=2x^2-1\) at \(x=3.\)
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15.

Find the value of function \(f(x)=x^4-5x^3+6\) at \(a=3.\)
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Hint.
\begin{equation*} \lim_{x\to a} f(x)= \lim_{x\to a} x^4-5x^3+6 = -38. \end{equation*}
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16.

Evaluate the function at \(a=3.\)
\begin{equation*} \lim_{x\to a} f(x)= \lim_{x\to a} \frac{x+1}{x-3} \end{equation*}
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17.

Evaluate:
\begin{equation*} \lim_{x\to 1} f(x)= \lim_{x\to 1} \left[-\frac{1}{(x-1)^2}\right] \end{equation*}
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18.

Find the relationship between \(a\) and \(b\) so that the function \(f\) defined by
\begin{equation*} f(x) = \begin{cases} ax+1; & \text{for } x\leq 3 \\ bx+1; & \text{for } x \gt 3 \\ \end{cases} \end{equation*}
is continuous at \(x=3.\)
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19.

Evaluate:
  1. \begin{equation*} \lim_{z\to 1} \frac{(5-z)^2-25}{z} \end{equation*}
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  2. \begin{equation*} \lim_{r\to 0} \frac{1}{r}\left(\frac{1}{r+3}-\frac{1}{3}\right) \end{equation*}
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  3. \begin{equation*} \lim_{r\to 1} \frac{\sqrt{x+3}-2}{x-1} \end{equation*}
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  4. \begin{equation*} \lim_{r\to -5} \frac{2x+10}{|x+5|} \end{equation*}
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    Hint.
    \begin{equation*} |x+5| = \begin{cases} x+5; & \text{for } x+5 \gt 0 \\ -(x+5); & \text{for } x+5 \lt 0 \\ \end{cases} \end{equation*}
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20.

Find all points of discontinuity of \(f\text{,}\) where
\begin{equation*} f(x) = \begin{cases} \frac{\sin x}{x} & x \lt 0 \\ x+1 & x \geq 0 \end{cases}. \end{equation*}
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21.

Examine the following funcitons for the continuity
  1. \begin{equation*} f(x)=x-5 \end{equation*}
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  2. \begin{equation*} f(x)=\frac{1}{x-5}; \, x\neq 5 \end{equation*}
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  3. \begin{equation*} f(x)=\frac{x^2-25}{x+5}; \, x\neq -5 \end{equation*}
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  4. \begin{equation*} f(x)=\lvert x-5 \rvert \end{equation*}
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22.

\begin{equation*} \lim_{x\to 5} \frac{\sqrt{4x+5}-5}{\sqrt{x-1}-2} \end{equation*}
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Answer: 8/5.
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23.

\begin{equation*} \lim_{x\to \infty} \frac{2x^2+x-5}{4x^3+5x+1} \end{equation*}
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24.

\begin{equation*} \lim_{x\to \infty} \frac{3x^5+5x^3+2}{2x^3+x-1} \end{equation*}
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25.

\begin{equation*} \lim_{x\to \infty} \frac{-2x^4+5x^2+x+1}{5x^3+x-2} \end{equation*}
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26.

Find:
\begin{equation*} \lim_{x\to 0} \frac{\sin 7x}{4x} \end{equation*}
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27.

Calculate:
  1. \begin{equation*} \lim_{x\to 0} x\cot x \end{equation*}
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  2. \begin{equation*} \lim_{x\to 0} \frac{\sin 3x}{5x^3-4x} \end{equation*}
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28.

Evaluate \(f(x)\text{:}\) If
  1. \begin{equation*} f(x) = \begin{cases} x^2-4; & x\lt 2 \\ 3x-2; & x\geq 2 \end{cases} \end{equation*}
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  2. \begin{equation*} f(x) = \begin{cases} x^2+3x; & x\lt 0 \\ 3x+4; & x\geq 0 \end{cases} \end{equation*}
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  3. \begin{equation*} f(x) = \begin{cases}2x+5; & x\lt -1 \\ -x+3; & -1\leq x \lt 2 \\ x^2-4x+7; & x \geq 2 \end{cases} \end{equation*}
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29.

Evaluate the following:
  1. \begin{equation*} \lim_{x\to \infty} \frac{3x^5+5x^3+2}{2x^3+x-1} \end{equation*}
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  2. \begin{equation*} \lim_{x\to -\infty} \frac{2x^3+x+1}{1-x-x^3} \end{equation*}
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  3. \begin{equation*} \lim_{x\to 2} \frac{x^2-3x+2}{x^2-4} \end{equation*}
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  4. \begin{equation*} \lim_{x\to 1} \frac{x^3-1}{x^6-1} \end{equation*}
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  5. \begin{equation*} \lim_{x\to 1} \frac{x^{\frac{1}{3}}-1}{x^2-1} \end{equation*}
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  6. \begin{equation*} \lim_{x\to 0} \frac{\sqrt{x+1} -1}{x } \end{equation*}
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  7. \begin{equation*} \lim_{x\to 0} \frac{\sin 3x}{x} \end{equation*}
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  8. \begin{equation*} \lim_{x\to 0} \frac{1-\cos 2x}{x^2} \end{equation*}
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  9. \begin{equation*} \lim_{x\to 0} \frac{\ln (1+x^2)}{1-\cos x} \end{equation*}
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  10. \begin{equation*} \lim_{x\to 0} \frac{\sin (\sin x)}{x} \end{equation*}
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  11. \begin{equation*} \lim_{x \to \frac{\pi}{2}} {\tan^{-1} \lfloor \cos x \rfloor} \end{equation*}
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30.

Find points of discontinuity if any in the following functions:
  1. \begin{equation*} f(x) = \begin{cases} x+1; & x\geq 1 \\ x^2+1; & x \lt 1 \end{cases} \end{equation*}
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  2. \begin{equation*} f(x) = \begin{cases} \frac{x}{|x|}; & x \lt 0 \\ -1; & x\geq 0 \end{cases} \end{equation*}
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31.

  1. Determine if \(f(x) = \frac{x^2-4}{x-2}\) is contunous at \(x=2.\)
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  2. Determine if \(g(x) = \frac{1}{x-3}\) is contunous at \(x=3.\)
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  3. Show that the function \(f(x) = x^2 \) is contunous.
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  4. Let
    \begin{equation*} f(x) = \begin{cases} \frac{1-\cos 4x}{x^2} & x \lt 0 \\ a & x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4} & x \gt 0 \end{cases} \end{equation*}
    Determine the value of \(a\) so that \(f(x)\) is continuous at \(x=0\text{.}\)
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  5. Let
    \begin{equation*} f(x) = \begin{cases} \frac{(e^x-1)^3\csc (ax)}{\ln (1+x^2)} & x \neq 0 \\ b & x=0 \end{cases} \end{equation*}
    is continuous, then prove \(ab =1\text{.}\)
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  6. Let
    \begin{equation*} f(x) = \begin{cases} (1+3x)^{1/x} & x \neq 0 \\ e^3 & x=0 \end{cases} \end{equation*}
    Discuss the continuity of \(f(x)\) at \(x=0\text{.}\)
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  7. Let
    \begin{equation*} f(x) = \begin{cases} (\cos x)^{\cot^2 x} & x \neq 0 \\ e^{-1/2} & x=0 \end{cases} \end{equation*}
    Find whether the function \(f(x)\) is continuous at \(x=0\text{.}\)
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Hint.
For \(1^\infty \) form, use
\begin{equation*} \lim_{x \to 0} {f(x)^{g(x)}} = e^{\lim_{x \to 0} (f(x)-1)\times g(x)} \end{equation*}
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32.

Evaluate:
\begin{equation*} \lim_{x\to 0} \frac{(e^{x^2} -1)\ln(1+\sin^2 x)}{(1-\cos 2 x)^2} \end{equation*}
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\begin{equation*} = \lim_{x\to 0} \frac{(e^{x^2} -1)\ln(1+\sin^2 x)}{(2\sin 2 x)^2} \end{equation*}
\begin{equation*} = \lim_{x\to 0} \frac{(e^{x^2} -1)}{(x)^2}\times \frac{\ln(1+\sin^2 x)}{\sin^2 x} \times \frac{x^2}{4\sin^2 x} = 1/4. \end{equation*}
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33.

Is \(f(x) = 5x^4 - 2x + 7\) continuous at \(x = -1\text{?}\)
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Answer: (Yes — polynomial, continuous everywhere.)
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34.

Determine if \(g(x) = \frac{1}{x-3}\) is continuous at \(x = 3\text{.}\)
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Answer: (No, undefined at \(x = 3\text{.}\))
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35.

Check continuity at \(x = 0\text{:}\)
\begin{equation*} h(x) = \begin{cases} x^2+1 & x \leq 0 \\ 2x+1 & x \gt 0 \end{cases} \end{equation*}
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Answer: (Yes, both sides \(\to 1,\) and \(h(0) = 1\text{.}\))
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36.

For what value of \(k\) is the function continuous at \(x = 2\text{?}\)
\begin{equation*} f(x) = \begin{cases} 3x-1 & x \lt 2 \\ k & x = 2 \\ x+4 & x \gt 2 \end{cases} \end{equation*}
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Hint.
(Set left limit = right limit: \(3(2)-1 = 5\text{,}\) \(2+4 = 6\) \(\rightarrow\) no \(k\) works. Discontinuous.)
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37.

Is \(f(x) = |x - 1|\) continuous at \(x = 1\text{?}\)
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Answer: (Yes,absolute value is continuous everywhere.)
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38.

Find and classify all discontinuities: \(f(x)=\frac{x^2-1}{x-1}\)
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Hint.
(Removable at \(x = 1\text{;}\) limit = 2. Define \(f(1) = 2\) to remove.)
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39.

Show that the function \(f(x) = \cos x^2\) is continuous.
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Hint.
Take a point \(x=c\) where the function is continuous. Then, at \(x=c,\) \(f(x)=f(c)=\cos c^2\)
\begin{equation*} \lim_{x\to c^-} f(x)= \lim_{h\to 0} \cos (c-h)^2 =\cos c^2. \end{equation*}
and
\begin{equation*} \lim_{x\to c^+} f(x)= \lim_{h\to 0} \cos (c+h)^2 =\cos c^2. \end{equation*}
i.e., the given function is continuous at \(x=c.\)
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40.

Discuss the continuity of
\begin{equation*} f(x)= \begin{cases} x^2+2 & x \lt 0 \\ 3x+2 & 0\leq x \lt 1 \\ 3e^x-2 & x\geq 1 \end{cases} \, \text{ at } x=0, x=1. \end{equation*}
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41.

Discuss the continuity of \(f(x) = |x-1| +|x-2| \) at \(x=1 \) and \(x=2.\)
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42.

Let \(\lfloor t \rfloor \) denote the greatest integer \(\leq t\) and \(\lim_{x\to 0} x\lfloor \frac{4}{x}\rfloor =A.\) The function \(f(x)= \lfloor x^2 \rfloor \sin \pi x\) is discontinuous when \(x=?\)
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Hint.
\(\lfloor x\rfloor + {x} =x.\)
\begin{equation*} \lim_{x\to 0} f(x)= \lim_{x\to 0} x\lfloor \frac{4}{x}\rfloor =A \end{equation*}
\begin{equation*} \lim_{x\to 0} x \left(\frac{4}{x}-{x}\right ) =A \end{equation*}
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43.

Discuss the continuity of the following functions
  1. \begin{equation*} f(x)= \begin{cases} 1 & \text { if } x \leq 0 \\ 2 & \text { if } x \gt 0 \end{cases} \end{equation*}
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  2. \begin{equation*} f(x)= \begin{cases} 1 & \text { if } x \neq 0 \\ 2 & \text { if } x = 0 \end{cases} \end{equation*}
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  3. \begin{equation*} f(x)= \begin{cases} -x & \text { if } x \lt 0 \\ x & \text { if } x \geq 0 \end{cases} \end{equation*}
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