Consider $\lim _ { x \rightarrow - 7 } \left( \frac { 2 x } { 7 } - 5 \right) = - 7$ In order for $\left| \left( \frac { 2 x } { 7 } - 5 \right) - ( - 7 ) \right|

Question

Consider $\lim _ { x \rightarrow - 7 } \left( \frac { 2 x } { 7 } - 5 \right) = - 7$ In order for $\left| \left( \frac { 2 x } { 7 } - 5 \right) - ( - 7 ) \right| < 0.01$ whenever $0 < | x - ( - 7 ) | < \delta$ which of the following is true for the largest ?
A)0.001
B)0.025
C)0.03
D)0.035
E)Impossible to find

Quizplus · Accepted Answer

To solve this, we simplify the given inequality: $\left| \left( \frac { 2 x } { 7 } - 5 \right) + 7 \right| < 0.01$. Simplifying inside the absolute value gives $\left| \frac { 2 x } { 7 } + 2 \right| < 0.01$. Further simplification leads to $\left| \frac { 2 (x + 7) } { 7 } \right| < 0.01$, which simplifies to $|x + 7| < 0.035$. Thus, $\delta = 0.035$ is the largest value that satisfies the condition, making option D correct.

Consider lim⁡x→−7(2x7−5)=−7\lim _ { x \rightarrow - 7 } \left( \frac { 2 x } { 7 } - 5 \right) = - 7limx→−7​(72x​−5)=−7

Consider $\lim _ { x \rightarrow - 7 } \left( \frac { 2 x } { 7 } - 5 \right) = - 7$