To determine whether a limit exists and estimate its value, you need to inspect the behavior of the function $f(x)$ as $x$ approaches specific values from both sides on the graph. Let's assume you are referring to a piecewise or continuous function, as these are typical in limit problems. Here's how to approach it:

(a) $\lim_{{x \to 7}} f(x)$

• Look at the value of $f(x)$ as $x$ approaches 7 from both the left-hand side ($x \to 7^-$) and the right-hand side ($x \to 7^+$).
• If the graph approaches the same value from both directions, then the limit exists.
• Does it exist?
  • If the left-hand and right-hand values are the same, the limit exists.
  • Estimate: If the function approaches a certain value (say $L$) near $x = 7$, then the limit is $L$.

(b) $\lim_{{x \to 6}} f(x)$

• Again, examine the graph at $x = 6$ from both sides.
• Does it exist?
  • Check if the left-hand and right-hand limits are equal. If they are, the limit exists.
  • Estimate: If the graph approaches a specific value near $x = 6$, the limit can be estimated as that value.

Conclusion

• If the graph shows a discontinuity (jump, vertical asymptote, or oscillation) near $x = 7$ or $x = 6$, the limit may not exist.
• If the function smoothly approaches the same value from both sides, the limit exists and can be estimated from the graph.

If you can upload a graph, I can give a more detailed answer. Would you like me to explain more about how limits behave around discontinuities or provide further clarification?

5 Related Questions:

1. What happens if the left-hand and right-hand limits do not match?
2. Can a function have a limit at a point where it is not defined?
3. What are the types of discontinuities that affect limits?
4. How can you estimate a limit if the graph approaches infinity?
5. How does the concept of one-sided limits help in limit problems?

Tip:

Always check both sides of the graph (left and right) when estimating limits to ensure they match for the limit to exist.