Questions: Example 6: Find the limits using algebra. Use numerical or graphical evidence to determine the left and right-hand limits of the function. lim x→4− 28-7x/(4-x)= lim x→4+ 28-7x/(4-x)= lim x→4 28-7x/(4-x)= Example 7: Find the limits using algebra. Evaluate the limit:

Solution Steps

1. For the left-hand limit as \( x \to 4^- \), evaluate the expression \(\frac{|28-7x|}{4-x}\) by considering values of \( x \) slightly less than 4. Since \( 28 - 7x \) becomes negative as \( x \) approaches 4 from the left, the absolute value will change the sign.
2. For the right-hand limit as \( x \to 4^+ \), evaluate the expression \(\frac{|28-7x|}{4-x}\) by considering values of \( x \) slightly greater than 4. Here, \( 28 - 7x \) remains positive, so the absolute value does not change the sign.
3. For the two-sided limit as \( x \to 4 \), compare the left-hand and right-hand limits. If they are equal, that is the limit; otherwise, the limit does not exist.

To find the left-hand limit as \( x \to 4^- \), we evaluate the expression \(\frac{|28 - 7x|}{4 - x}\) for values of \( x \) slightly less than 4. In this case, \( 28 - 7x \) becomes negative, so the absolute value changes the sign:

\[ \lim_{x \to 4^-} \frac{|28 - 7x|}{4 - x} = \lim_{x \to 4^-} \frac{-(28 - 7x)}{4 - x} = \lim_{x \to 4^-} \frac{7x - 28}{4 - x} = 7 \]

For the right-hand limit as \( x \to 4^+ \), we evaluate the expression \(\frac{|28 - 7x|}{4 - x}\) for values of \( x \) slightly greater than 4. Here, \( 28 - 7x \) remains positive, so the absolute value does not change the sign:

\[ \lim_{x \to 4^+} \frac{|28 - 7x|}{4 - x} = \lim_{x \to 4^+} \frac{28 - 7x}{4 - x} = -7 \]

The two-sided limit as \( x \to 4 \) is determined by comparing the left-hand and right-hand limits. Since the left-hand limit is 7 and the right-hand limit is -7, the two-sided limit does not exist:

\[ \lim_{x \to 4} \frac{|28 - 7x|}{4 - x} \text{ does not exist} \]

Final Answer