Approaching a Number: Introduction to Limits
Trigonometry - Approaching a Number (Limit) - How it Works - Video
Example 1
Example 1:
When we approach a number, we use the x-values and then use the y-value for our answer. When we approach a number we have two ways to do that: one from the left side of the graph and one from the right side of the graph. The reason that we say that is if a number is in Q II or Q III, we have negative numbers on the right side so it might get confusing if we say from the positive numbers or the negative numbers.
In this example, we have the graph y = sinθ. The first number that we want to approach is π. We have two sides: the numbers to the left, the orange line, and the numbers to the right, the blue line.
When we approach π from the left, the orange line, we are getting closer and closer until we reach the point (π , 0). Since the y-value is our answer, we write as x → π -, sinθ → 0 (as x approaches π from the left, sinθ yields 0).
When we approach π from the right, the blue line, we are getting closer and closer until we reach the point (π , 0). Since the y-value is our answer, we write as x → π +, sinθ → 0 (as x approaches π from the right, sinθ yields 0)
Example 1 continued:
We are still working with the graph y = sinθ. The second number that we want to approach is π / 2. We have two sides: the numbers to the left, the orange line, and the numbers to the right, the blue line.
When we approach π from the left, the orange line, we are getting closer and closer until we reach the point (π / 2 , 1). Since the y-value is our answer, we write as x → (π / 2)-, sinθ → 1 (as x approaches π/2 from the left, sinθ yields 1).
When we approach π from the right, the blue line, we are getting closer and closer until we reach the point (π / 2, 1). Since the y-value is our answer, we write as x → (π / 2)+, sinθ → 1 (as x approaches π/2 from the right, sinθ yields 1).
Example 2
Example 2:
When we approach a number, we use the x-values and then use the y-value for our answer. When we approach a number we have two ways to do that: one from the left side of the graph and one from the right side of the graph. The reason that we say that is if a number is in Q II or Q III, we have negative numbers on the right side so it might get confusing if we say from the positive numbers or the negative numbers.
In this example, we have the graph y = cscθ. The first number that we want to approach is π. We have two sides: the numbers to the left, the orange line, and the numbers to the right, the green line.
When we approach π from the left, the orange line, we are getting closer and closer until we reach the vertical asymptote at x = π. Remember the y-value is our answer. Do we have any y-value with a vertical asymptote? No, because csc(π) is undefined. Since the graph never reaches π, it goes on forever and forever without touching the vertical asymptote or to ± infinity.
When we approach π from the left, the orange line, we are getting closer and closer until we reach the vertical asymptote at x = π . Since the y-value is undefined, we write as x → π -, cscθ → +∞ (as x approaches π from the left, cscθ yields +∞).
When we approach π from the right, the orange line, we are getting closer and closer until we reach the vertical asymptote at x = π . Since the y-value is undefined, we write as x → π +, cscθ → -∞ (as x approaches π from the left, cscθ yields -∞).
Example 2 continued:
We are still working with the graph y = cscθ. The second number that we want to approach is 3π / 2. We have two sides: the numbers to the left, the orange line, and the numbers to the right, the green line.
When we approach π from the left, the orange line, we are getting closer and closer until we reach the point (3π / 2 , -1). Since the y-value is our answer, we write as x → (3π / 2)-, cscθ → -1 (as x approaches π/2 from the left, cscθ yields -1).
When we approach π from the right, the blue line, we are getting closer and closer until we reach the point (3π / 2, -1). Since the y-value is our answer, we write as x → (3π / 2)+, cscθ → -1 (as x approaches π/2 from the right, cscθ yields -1).