WHAT ARE TRIGONOMETRIC LIMITS? (EXERCISES SOLVED) - MATHS - 2025

The trigonometric limits are limits of functions such that these functions are formed by trigonometric functions.

There are two definitions that must be known in order to understand how to calculate a trigonometric limit.

These definitions are:

- Limit of a function «f» when «x» tends to «b»: it consists in calculating the value to which f (x) approaches as «x» approaches «b», without reaching «b» ».

- Trigonometric functions: the trigonometric functions are the sine, cosine and tangent functions, denoted by sin (x), cos (x) and tan (x) respectively.

The other trigonometric functions are obtained from the three functions mentioned above.

Function limits

To clarify the concept of a function limit, we will proceed to show some examples with simple functions.

- The limit of f (x) = 3 when "x" tends to "8" is equal to "3", since the function is always constant. No matter how much "x" is worth, the value of f (x) will always be "3".

- The limit of f (x) = x-2 when «x» tends to «6» is «4». Since when "x" approaches "6" then "x-2" approaches "6-2 = 4".

- The limit of g (x) = x² when "x" tends to "3" is equal to 9, since when "x" approaches "3" then "x²" approaches "3² = 9".

As can be seen in the previous examples, calculating a limit consists of evaluating the value to which "x" tends in the function, and the result will be the value of the limit, although this is true only for continuous functions.

Are there more complicated limits?

The answer is yes. The above examples are the simplest examples of limits. In calculus books, the main limit exercises are those that generate an indeterminacy of the type 0/0, ∞ / ∞, ∞-∞, 0 * ∞, (1) ^ ∞, (0) ^ 0 and (∞) ^ 0.

These expressions are called indeterminacies since they are expressions that do not make sense mathematically.

In addition, depending on the functions involved in the original limit, the result obtained when solving the indeterminacies may be different in each case.

Examples of Simple Trigonometric Limits

To solve limits, it is always very useful to know the graphs of the functions involved. The graphs of the sine, cosine, and tangent functions are shown below.

Some examples of simple trigonometric limits are:

- Calculate the limit of sin (x) when «x» tends to «0».

When looking at the graph it can be seen that if "x" gets closer to "0" (both from the left and right), then the sine graph also gets closer to "0". Therefore, the limit of sin (x) when "x" tends to "0" is "0".

- Calculate the limit of cos (x) when «x» tends to «0».

Observing the graph of the cosine it can be seen that when "x" is close to "0" then the graph of the cosine is close to "1". This implies that the limit of cos (x) when "x" tends to "0" is equal to "1".

A limit can exist (be a number), as in the previous examples, but it can also happen that it does not exist as shown in the following example.

- The limit of tan (x) when «x» tends to «Π / 2» from the left is equal to «+ ∞», as can be seen in the graph. On the other hand, the limit of tan (x) when "x" tends to "-Π / 2" from the right is equal to "-∞".

Trigonometric Limits Identities

Two very useful identities when calculating trigonometric limits are:

- The limit of «sin (x) / x» when «x» tends to «0» is equal to «1».

- The limit of «(1-cos (x)) / x» when «x» tends to «0» is equal to «0».

These identities are used very often when you have some kind of indeterminacy.

Solved exercises

Solve for the following limits using the identities described above.

- Calculate the limit of «f (x) = sin (3x) / x» when «x» tends to «0».

If the function "f" is evaluated at "0", an indeterminacy of type 0/0 will be obtained. Therefore, we must try to solve this indeterminacy using the identities described.

The only difference between this limit and the identity is the number 3 that appears within the sine function. In order to apply the identity, the function «f (x)» must be rewritten in the following way «3 * (sin (3x) / 3x)». Now both the sine argument and the denominator are equal.

So when "x" tends to "0", using the identity gives "3 * 1 = 3". Therefore, the limit of f (x) when "x" tends to "0" is equal to "3".

- Calculate the limit of «g (x) = 1 / x - cos (x) / x» when «x» tends to «0».

When "x = 0" is substituted in g (x), an indeterminacy of the type ∞-∞ is obtained. To solve it, the fractions are first subtracted, which yields "(1-cos (x)) / x".

Now, applying the second trigonometric identity we have that the limit of g (x) when «x» tends to «0» is equal to 0.

- Calculate the limit of «h (x) = 4tan (5x) / 5x» when «x» tends to «0».

Again, if h (x) is evaluated at "0", an indeterminacy of type 0/0 will be obtained.

Rewriting as (5x) as sin (5x) / cos (5x) results in h (x) = (sin (5x) / 5x) * (4 / cos (x)).

Using that the limit of 4 / cos (x) when "x" tends to "0" is equal to "4/1 = 4" and the first trigonometric identity is obtained that the limit of h (x) when "x" tends a "0" is equal to "1 * 4 = 4".

Observation

Trigonometric limits are not always easy to solve. Only basic examples were shown in this article.

References

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