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).

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