Limits of Trigonometric Functions:- Limits are also applied to trigonometry functions along with the general ones. In geometric figures, to calculate unknown angles and distances from known or measured angles, we have to use trigonometric functions.

Limits cannot be applied to all functions because x approaches infinity. Let the function f(x) = x cos(x), as** x** raises larger, this function does not approach any specific real number meanwhile we can always choose a value of **x** to make f(x) greater than whatever number we want.

**What are Trigonometric functions?**

The word trigonometry comes from the Greek language and is derived from a combination of three Greek words. The three Greek words from which trigonometry is derived are Trei (three), Goni (angles), and Metron (measures) refer to triangle measurements.

There are six functions of an angle that are frequently used in trigonometry. These functions are derived by a triangle having a base, perpendicular, and hypotenuse.

Sine is found by using a right-angled triangle.

Sine = perp / hypotenuse

Cosine is found by using a right-angled triangle.

Cosine = altitude / hypotenuse

Tangent is found by using a right-angled triangle.

Tangent = perpendicular / altitude

We can also find tangent by sine and cosine due to relationship.

Tangent = sine / cosine

**What are limits?**

Mathematical limits are real numbers that are unique. Consider the limit of a real-valued function “f” and a real number “c,” which is written as.

f(x) = L

As x approaches **an** equal to L, it can be read as the limit of f(x).

The term **one-sided limit** refers to a restriction that occurs only on one side, either left or right.

A **two-sided limit** is one that has a limit on both sides.

The term **infinite limit** refers to a function that can increase or decrease indefinitely.

**Example **

Evaluate 4x^{2} + 9x + 5 / 16x^{2} – 25

**Solution**

**Step 1:** Write the given limit function.

4x^{2} + 9x + 5 / 16x^{2} – 25

**Step 2:** Make factors of the numerator by factorization method.

4x^{2} + 9x + 5 / 16x^{2} – 25 = 4x^{2} + 4x + 5x + 5 / 16x^{2} – 25

4x^{2} + 9x + 5 / 16x^{2} – 25 = (4x + 5) (x + 1) / 16x^{2} – 25

**Step 3**: Make factors of denominator with the help of formula.

4x^{2} + 9x + 5 / 16x^{2} – 25 = (4x + 5) (x + 1) / (4x + 5) (4x – 5)

**Step 4:** Now simplify the above equation.

= (4x + 5) (x + 1) / (4x + 5) (4x – 5)

= (x + 1) / (4x – 5)

**Step 5:** Apply the quotient of the function rule of limit.

= (x + 1) / (4x – 5)

**Step 6:** Apply limits:

4x^{2} + 9x + 5 / 16x^{2} – 25 = (5 + 1) / (4(5) – 5)

4x^{2} + 9x + 5 / 16x^{2} – 25 = 6 / (20 – 5) = 6/15 = 2/5

Limits are also applied to calculate the slope. The slope is a very common concept used widely in algebra. Slope is very essential for the calculation of the equation of the line.

**Rules of Limits**

Name |
Rules |

Constant rule | A = A |

Constant time’s function | (kfx) = k (fx) |

Difference of the functions Rule | x→c (f(x) – g(x)) = f(x) – g(x) |

Product of the functions Rule | x→c (f(x) * g(x)) = f(x) * g(x) |

The quotient of the functions Rule | x→c (f(x) / g(x)) = f(x) / g(x) |

**How to apply limits on trigonometric functions?**

The sine and cosine trigonometric functions have four essential limit properties. These principles can be used to solve a variety of limit problems utilizing the six fundamental trigonometric functions. Trigonometric limits can be easily evaluated by using the limit calculator. Three properties of limits are mentioned below.

- 1 – cos(x) / x = 0
- sin(x) / x = 1

- sec(x) – 1 / x = 0

**Example 1:**

Evaluate sin(8x) / x.

**Solution **

**Step 1:** write the given value.

sin(8x) / x

**Step 2:** Multiply and divide by 8.

8sin(8x) / 8x

**Step 3:** Apply the product of the function rule.

8 x→0 sin(8x) / 8x

**Step 4:** By constant rule and trigonometric limit property we have,

8 x→0 sin(8x) / 8x = 8 x 1 = 8

**Step 5:** Write the given input with result.

sin(8x) / x = 8

**Example 2:**

Evaluate sin(2x) / x.

**Solution **

**Step 1:** write the given value.

sin(2x) / x

**Step 2:** Multiply and divide by 2.

2sin(2x) / 2x

**Step 3:** Apply the product of the functions rule.

2 sin(2x) / 2x

**Step 4:** By a property, we can apply this.

2 sin(2x) / 2x = 2 x 0

**Step 5:** Write the given input with result.

sin(2x) / x = 0

**Example 3:**

Evaluate 2x^{2} + 4 / 2x^{2} -7x -4

**Solution **

**Step 1:** write the given value.

2x^{2} + 4 / 2x^{2} -7x -4

**Step 2:** Take x^{2} common from numerator and denominator.

x→ x^{2}(2 + 4 / x^{2}) / x^{2}(2 – 7/x – 4/x^{2})

**Step 3:** Cancel x^{2}.

x→ (2 + 4 / x^{2}) / (2 – 7/x – 4/x^{2})

**Step 4:** Apply limits.

2 + 4/∞^{2} / 2 – 7/∞ – 4/∞^{2}

**Step 5:** As n/∞ = 0, put zero where infinity is at the denominator.

2 + 0 / 2 – 0 – 0 = 2 / 2 = 1

**Step 6:** Write the given input with result.

2x^{2} + 4 / 2x^{2} -7x -4 = 1

**Example 4:**

Evaluate 2x^{3} – 3x^{2} + 4 / x^{3} -7x -4

**Solution **

**Step 1:** write the given value.

2x^{3} – 3x^{2} + 4 / x^{3} -7x -4

**Step 2:** Take x^{3} common from numerator and denominator.

x→ x^{3}(2 – 3/x + 4 / x^{3}) / x^{3}(1 – 7/x^{2} – 4/x^{3})

**Step 3:** Cancel x^{2}.

x→ (2 – 3/x + 4 / x^{3}) / (1 – 7/x^{2} – 4/x^{3})

**Step 4:** Apply limits.

2 – 3/∞ + 4/∞^{3} / (1 – 7/∞^{2} – 4/∞^{3}

**Step 5:** As n/∞ = 0, put zero where infinity is at the denominator.

2 – 0 + 0 / 1 – 0 – 0 = 2 / 1 = 2

**Step 6:** Write the given input with result.

2x^{3} – 3x^{2} + 4 / x^{3} -7x -4 = 2

**Relationship of slope of a tangent line and limits **

Limits are also applied to calculate the slope. The slope is a very common concept used widely in algebra. It is very essential for the calculation of the equation of the line.

As derivatives are used to calculate the slope of the tangent, the limits are also helpful in this regard. Because limits are used to calculate the derivative which is the slope of the tangent. You can use an online tool like slope calculator to find the slope of line, slope-intercept form, and point slope form.

**Summary **

The limits of trigonometric functions can also be calculated by using the above-mentioned properties. Evaluating the limits of trigonometric functions is not much difficult, just a little effort is required to master the concept.