Name 1.6 Limits and continuity Homework Date Period 1

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Name 1.6 Limits and continuity Homework Date Period 1. Use the graph of the function y = g(x) shown below, to evaluate each of the following. A. lim g(x) = B. lim 9(x) = ya Clim, g(x) = D. 9(-1) = ad E. lim g(x) = Elim, g(x) = . Glim g(x) = H. lim g(x) = J. g(7) = K. lim g(x) = 2. Use the graph of the function y = f(x) shown

The correct answer and explanation is:

To answer your homework questions, let’s first define the basic concepts of limits and continuity in calculus:

1. Limit Definition:

The limit of a function f(x)f(x) as xx approaches a value aa is the value that f(x)f(x) gets closer to as xx gets closer to aa. In mathematical notation: lim⁡x→af(x)=L\lim_{{x \to a}} f(x) = L

if for every small value ϵ>0\epsilon > 0, there is a small value δ>0\delta > 0 such that for all xx within the range (a−δ,a+δ)(a – \delta, a + \delta), f(x)f(x) stays within the range (L−ϵ,L+ϵ)(L – \epsilon, L + \epsilon).

2. Continuity:

A function f(x)f(x) is continuous at a point aa if:

  • f(a)f(a) exists.
  • The limit lim⁡x→af(x)\lim_{{x \to a}} f(x) exists.
  • lim⁡x→af(x)=f(a)\lim_{{x \to a}} f(x) = f(a).

Now let’s proceed with the evaluation of limits and function values using the graph of g(x)g(x) and f(x)f(x). I would need to see the graph of these functions in order to give precise answers. However, I can still provide a general approach for how you can evaluate these limits from the graph.

Example Evaluation Based on the Graph:

  1. A. lim⁡x→ag(x)\lim_{{x \to a}} g(x): Look at the graph near x=ax = a. If the graph approaches a particular value from both sides of x=ax = a, then that value is the limit of g(x)g(x) as x→ax \to a.
  2. B. lim⁡x→a+g(x)\lim_{{x \to a^+}} g(x): This is the limit of g(x)g(x) as xx approaches aa from the right (from values larger than aa). Look at the graph from the right side of x=ax = a to find this limit.
  3. C. lim⁡x→a−g(x)\lim_{{x \to a^-}} g(x): This is the limit of g(x)g(x) as xx approaches aa from the left (from values smaller than aa). Again, look at the graph from the left side of x=ax = a to evaluate this limit.
  4. D. g(−1)g(-1): To evaluate g(−1)g(-1), simply locate the point on the graph where x=−1x = -1. The corresponding value of yy at x=−1x = -1 is g(−1)g(-1).
  5. E. lim⁡x→bg(x)\lim_{{x \to b}} g(x): Similar to the first step, but look at x=bx = b this time. The limit will be the value the function is approaching from both sides.
  6. F. lim⁡x→b+g(x)\lim_{{x \to b^+}} g(x) and G. lim⁡x→b−g(x)\lim_{{x \to b^-}} g(x): These are the right-hand and left-hand limits, respectively. Use the graph to analyze the function’s behavior from the right and left of x=bx = b.
  7. H. lim⁡x→cg(x)\lim_{{x \to c}} g(x): Look at the graph near x=cx = c. If the function is approaching a specific value from both directions, that’s your answer.
  8. I. g(7)g(7): Find the value of the function at x=7x = 7. Look for the point where x=7x = 7 on the graph and note the corresponding value of g(7)g(7).
  9. J. lim⁡x→dg(x)\lim_{{x \to d}} g(x): Evaluate the limit at x=dx = d similarly to previous steps. If the graph shows a jump or discontinuity at dd, consider if the left-hand limit and right-hand limit are equal.
  10. K. lim⁡x→eg(x)\lim_{{x \to e}} g(x): Evaluate the limit at x=ex = e. If the graph approaches a specific value from both directions, that’s your limit.

Explanation:

  • Continuity: If the left-hand limit, right-hand limit, and the function’s value at a specific point are all equal, then the function is continuous at that point.
  • Discontinuities: If there is a jump or the left-hand and right-hand limits do not match, the function is discontinuous at that point.

These general steps should help you evaluate limits and function values directly from the graph.

If you can provide the graph or describe it in more detail, I can give you specific answers based on the given functions.

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