Jul 11, 2024

**Limit:**A value that a function approaches as the input approaches some value.**Direct Substitution:**Plugging in the value of`x`

directly into the function to find the limit.**Indeterminate Forms:**Forms like`0/0`

that need further analysis to evaluate.

**Limit as x approaches 2 of (x^2 - 4)/(x - 2):**

- Direct substitution: Results in
`0/0`

which is undefined. - Try values close to 2 (e.g.,
`2.1`

): The limit approaches 4. - Factorization method:
`(x^2 - 4) = (x + 2)(x - 2)`

.- Cancel out
`(x - 2)`

. - Evaluate remaining expression
`x + 2`

: Result is 4.

- Cancel out

**Limit as x approaches 5 of x^2 + 2x - 4:**

- Direct substitution works: Evaluate as
`5^2 + 2(5) - 4`

= 31.

**Limit as x approaches 3 of (x^3 - 27)/(x - 3):**

- Direct substitution gives
`0/0`

. - Factorize
`x^3 - 27`

: Use the difference of cubes formula.`(x^3 - 27) = (x - 3)(x^2 + 3x + 9)`

.- Cancel out
`(x - 3)`

and evaluate remaining expression: Result is 27.

**Limit as x approaches 3 of 1/x - 1/3 / (x - 3):**

- Multiply by the common denominator
`3x`

on top and bottom. - Simplify and cancel terms.
- Direct substitution: Result is
`-1/9`

.

**Limit as x approaches 9 of (sqrt(x) - 3) / (x - 9):**

- Multiply by the conjugate
`sqrt(x) + 3`

. - Simplify and cancel terms.
- Direct substitution: Result is
`1/6`

.

**Limit as x approaches 4 of 1/sqrt(x) - 1/2 / (x - 4):**

- Multiply by
`2sqrt(x)`

and by the conjugate. - Factor out
`-1`

and simplify. - Direct substitution: Result is
`-1/16`

.

- Identify
`x`

value and approach it from the left/right on the graph. - The limit is the
`y`

value the function approaches. - If left and right limits are not the same, the limit does not exist.

**Limit as x approaches -3:**

- From the left:
`y`

approaches 1. - From the right:
`y`

approaches -3. - Conclusion: Limit does not exist as the values do not match.

- One-sided limits (values approaching from left or right).
- Function values at specific points (closed circles).

**Limit as x approaches -2:**

- From the left and right:
`y`

value is -2. - Middle limit exists and equals -2.
- Function value at -2:
`y`

is 2 (based on closed circle).

**Limit as x approaches 1:**

- From the left:
`y`

value is 1. - From the right:
`y`

value is 3. - Middle limit does not exist as values do not match.
- Function value at 1:
`y`

is 2 (closed circle).

**Limit as x approaches 3:**

- As
`x`

approaches 3: Exhibit a vertical asymptote.- From the left:
`y`

goes to negative infinity. - From the right:
`y`

goes to positive infinity.

- From the left:
- Middle limit does not exist. Function value is undefined.

**Jump Discontinuity:**Non-removable; graph “jumps” at a specific point.**Hole:**Removable discontinuity; small gap in the graph.**Infinite Discontinuity:**Caused by vertical asymptotes; non-removable.