Limits and continuity

​Squeeze (Sandwich) Theorem

By Mr. Bivash Chakraborty

Topic - Limits and Continuity

By Mr. Bivash Chakraborty

But first, a quick refresher on limits!

Multiple Choice

Find the limit of $$\lim_{x\rightarrow0}\ f\left(x\right)$$ .

• 0

• Does not exist (DNE)

• -1

• $$\infty$$

• 5

Multiple Choice

Find the limit of $$\lim_{x\rightarrow-\frac{1}{2}}\ f\left(x\right)$$ .

• 0

• Does not exist (DNE)

• -1

• $$\infty$$

• 5

Multiple Choice

Find the limit of $$\lim_{x\rightarrow-1^-}\ f\left(x\right)\ and\ \lim_{x\rightarrow-1^+}\ f\left(x\right)$$ .

• Does not exist (DNE)

• 0, -7

• -2, 0

• -1, -7

• -2, -7

Multiple Choice

Find the limit of $$\lim_{x\rightarrow0}\ f\left(x\right)$$ .

• Does not exist (DNE)

• 1

• -1

• 1, -1

• 0

Multiple Choice

Find the limit of $$\lim_{x\rightarrow-1}\ \frac{\left(\sqrt[]{x\ +\ 10}\ -\ 3\right)}{x\ +\ 1}$$ .

• 0

• Does not exist (DNE)

• -1

• 1/6

• 6

Multiple Choice

Find the limit of $$\lim_{x\rightarrow0}\ \ \sqrt[4]{2\cos\left(x\right)-5}\$$ .

• 0

• Does not exist (DNE)

• $$-\infty$$

• $$+\infty$$

• 1

Multiple Choice

Find the limit of $$\lim_{x\rightarrow1^-}\ \ \ \frac{1}{x\ -\ 1}+e^{x^2}$$ .

• 0

• Does not exist (DNE)

• $$-\infty$$

• $$+\infty$$

• 1

Multiple Choice

Find the limit of $$\lim_{x\rightarrow0^+}\ \ \ \sin\frac{1}{x\ }$$ .

• 0

• Does not exist (DNE)

• $$-\infty$$

• $$+\infty$$

• 1

Multiple Choice

Find the limit of $$\lim_{x\rightarrow0^+}\ \ \ x\ \sin\frac{1}{x\ }$$ .

• 0

• Does not exist (DNE)

• $$-\infty$$

• $$+\infty$$

• 1

Multiple Choice

Find the limit of $$\lim_{x\rightarrow0^{ }}\ \ \ x^2\ \sin\frac{1}{x\ }$$ .

• 0

• Does not exist (DNE)

• $$-\infty$$

• $$+\infty$$

• 1

By Mr. Bivash Chakraborty

Citations: https://calcworkshop.com/limits/squeeze-theorem/

The Squeeze
(sandwich)
Theorem!!

By Mr. Bivash Chakraborty

lets take a look at a few examples:

By Mr. Bivash Chakraborty

​There's 2 special Trig limits...

​Never forget them. We will discuss them in the class in detail!

​proof: https://web.ma.utexas.edu/users/ikmartin/pages/writing/blog/calc_squeeze/#area_of_the_large_triangle

By Mr. Bivash Chakraborty

Moving on!

By Mr. Bivash Chakraborty

What is continuity?!

By Mr. Bivash Chakraborty

Some properties of continuity

By Mr. Bivash Chakraborty

Multiple Choice

Where is the graph discontinuous?

• -2, 0, 3

• 0, 3

• -2 only

• 2, 1, 0

• 3 only

By Mr. Bivash Chakraborty

Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are

continuous on all points in their domains.

By Mr. Bivash Chakraborty

Intermediate
Value
Theorem

By Mr. Bivash Chakraborty

Formally...