# 13 Lecture

## MTH101

### Midterm & Final Term Short Notes

## Limits and Continuity of Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are fundamental functions that appear in various areas of mathematics, science, and engineering.

**Important Mcq's**

Midterm & Finalterm Prepration

Past papers included

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**What is the limit of the sine function as x approaches infinity?**

a) 0

b) 1

c) does not exist

d) -1

Answer: c) does not exist

**What is the limit of the cosine function as x approaches ?/2?**

a) 0

b) 1

c) does not exist

d) -1

Answer: c) does not exist

**What is the derivative of the function f(x) = cos(x) - 2sin(x)?**

a) -cos(x) - 2cos(x)

b) -sin(x) - 2cos(x)

c) sin(x) - 2cos(x)

d) -sin(x) + 2cos(x)

Answer: b) -sin(x) - 2cos(x)

**Which of the following trigonometric functions has a vertical asymptote at x = ?/2?**

a) sine

b) cosine

c) tangent

d) none of the above

Answer: c) tangent

**What is the limit of the tangent function as x approaches ?/2 from the left?**

a) -?

b) ?

c) does not exist

d) 0

Answer: a) -?

**Which of the following trigonometric functions is continuous on the entire real line?**

a) sine

b) cosine

c) tangent

d) none of the above

Answer: d) none of the above

**What is the derivative of the function f(x) = sin(x)cos(x)?**

a) cos^2(x)

b) -cos^2(x)

c) 2sin(x)cos(x)

d) -2sin(x)cos(x)

Answer: c) 2sin(x)cos(x)

**Which of the following functions is not continuous at x = 0?**

a) sin(x)/x

b) cos(x)/x

c) tan(x)/x

d) all of the above are continuous at x = 0

Answer: c) tan(x)/x

**What is the limit of the function f(x) = sin(1/x) as x approaches 0?**

a) 0

b) does not exist

c) 1

d) -1

Answer: b) does not exist

**What is the maximum value of the function f(x) = 2sin(x) + 3cos(x) on the interval [0, 2?]?**

a) 5

b) -5

c) 2

d) 3

Answer: a) 5

**Subjective Short Notes**

Midterm & Finalterm Prepration

Past papers included

Download PDF
What is the definition of the sine function?

Answer: The sine function is defined as the y-coordinate of a point on the unit circle in the coordinate plane.

Is the limit of the sine function as x approaches zero defined? Why or why not?

Answer: No, the limit of the sine function as x approaches zero is not defined because the function oscillates between -1 and 1 as x approaches zero.

What is the limit of the cosine function as x approaches zero?

Answer: The limit of the cosine function as x approaches zero is 1.

What is the definition of continuity?

Answer: A function is said to be continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point.

Is the tangent function continuous at all points? Why or why not?

Answer: No, the tangent function is not continuous at certain points where it has vertical asymptotes.

What is the derivative of the sine function?

Answer: The derivative of the sine function is the cosine function.

What is the derivative of the cosine function?

Answer: The derivative of the cosine function is the negative sine function.

What is the derivative of the tangent function?

Answer: The derivative of the tangent function is the secant squared function.

How can the continuity of trigonometric functions be used to solve problems in calculus?

Answer: The continuity of trigonometric functions can be used to find critical points and solve optimization problems.

What is the maximum value of the function f(x) = sin(x) + cos(x) on the interval [0, 2?]?

Answer: The maximum value of the function f(x) = sin(x) + cos(x) on the interval [0, 2?] is 2, which occurs at x = ?/4 and 9?/4.