a) Evaluate : 8.173 4.938 [3]634.4 using log tables. b) Evaluate if exist : x 0 x+4 2x

a) Evaluate: $\frac{8.173\times 4.938}{\sqrt[3]{634.4}}$ using log tables

Step 1: Break down the expression using logarithm properties

The expression is of the form $\frac{A\times B}{\sqrt[3]{C}}$, where:

• $A=8.173$
• $B=4.938$
• $C=634.4$

Using logarithm properties: $$\log \left(\frac{A\times B}{\sqrt[3]{C}}\right)=\log (A\times B)-\log (\sqrt[3]{C})$$ $$=\log A+\log B-\frac{1}{3}\log C$$

Step 2: Find the logarithms of $A$, $B$, and $C$ using log tables

We use common logarithm (base 10) tables.

1. For $A=8.173$:

   • From log tables, $\log 8.17=0.9121$
   • Correction for the remaining digits (0.003): $$\log 8.173=0.9121+(0.003\times 0.0043)=0.9121+0.0000129\approx 0.9121$$ (Note: The correction is negligible here, so we take $\log 8.173\approx 0.9121$)

2. For $B=4.938$:

   • From log tables, $\log 4.93=0.6928$
   • Correction for the remaining digits (0.008): $$\log 4.938=0.6928+(0.008\times 0.0043)=0.6928+0.0000344\approx 0.6928$$ (Again, negligible correction, so $\log 4.938\approx 0.6928$)

3. For $C=634.4$:

   • First, express $634.4$ in scientific notation: $6.344\times 10^2$
   • $\log 6.344$ (from log tables):
     • $\log 6.34=0.8021$
     • Correction for 0.004: $$\log 6.344=0.8021+(0.004\times 0.0043)=0.8021+0.0000172\approx 0.8021$$
     • Now, add the exponent: $$\log 634.4=\log (6.344\times 10^2)=\log 6.344+2=0.8021+2=2.8021$$

Step 3: Compute $\log (\text{Numerator})$ and $\log (\text{Denominator})$

• Numerator ($A\times B$): $$\log (A\times B)=\log A+\log B=0.9121+0.6928=1.6049$$

• Denominator ($\sqrt[3]{C}$): $$\log (\sqrt[3]{C})=\frac{1}{3}\log C=\frac{1}{3}\times 2.8021=0.9340$$

Step 4: Compute the final logarithm

$$\log \left(\frac{A\times B}{\sqrt[3]{C}}\right)=1.6049-0.9340=0.6709$$

Step 5: Find the antilogarithm to get the final answer

We need to find $x$ such that $\log x=0.6709$. From antilog tables:

• The characteristic is 0, so the number lies between 1 and 10.
• The mantissa 0.6709 corresponds to 4.685 (from antilog tables).

Thus: $$x=4.685$$

Final Answer: $$\boxed{4.685}$$

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b) Evaluate if it exists: ${\lim }_{x\to 0}\frac{\sqrt{x+4}-2}{x}$

Step 1: Direct substitution check

Substitute $x=0$: $$\frac{\sqrt{0+4}-2}{0}=\frac{2-2}{0}=\frac{0}{0}$$ This is an indeterminate form, so we must simplify the expression.

Step 2: Rationalize the numerator

Multiply the numerator and denominator by the conjugate of the numerator ($\sqrt{x+4}+2$): $$\frac{\sqrt{x+4}-2}{x}\times \frac{\sqrt{x+4}+2}{\sqrt{x+4}+2}=\frac{(\sqrt{x+4}{)}^2-(2{)}^2}{x(\sqrt{x+4}+2)}$$ $$=\frac{(x+4)-4}{x(\sqrt{x+4}+2)}=\frac{x}{x(\sqrt{x+4}+2)}$$

Step 3: Simplify the expression

Cancel $x$ in the numerator and denominator (valid since $x\ne 0$): $$\frac{1}{\sqrt{x+4}+2}$$

Step 4: Take the limit as $x\to 0$

$$\underset{x\to 0}{\lim }\frac{1}{\sqrt{x+4}+2}=\frac{1}{\sqrt{0+4}+2}=\frac{1}{2+2}=\frac{1}{4}$$

Final Answer: $$\boxed{\frac{1}{4}}$$