Calculus Examples

f (x) = {\begin{cases} 4 e^{x - 2} + a x - 3 a & x < 2 \\ x^{3} + a x^{2} + 5 & x \geq 2 \end{cases}

$f (x) = {\begin{cases} 4 e^{x - 2} + a x - 3 a & x < 2 \\ x^{3} + a x^{2} + 5 & x \geq 2 \end{cases}$

Step 1

Find the limit of

4 e^{x - 2} + a x - 3 a

$4 e^{x - 2} + a x - 3 a$ as

x

$x$ approaches

2

$2$ from the left.

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Step 1.1

Change the two-sided limit into a left sided limit.

lim_{x \to 2^{-}} 4 e^{x - 2} + a x - 3 a

$lim_{x \to 2^{-}} 4 e^{x - 2} + a x - 3 a$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as

x

$x$ approaches

2

$2$ .

lim_{x \to 2^{-}} 4 e^{x - 2} + lim_{x \to 2^{-}} a x - lim_{x \to 2^{-}} 3 a

$lim_{x \to 2^{-}} 4 e^{x - 2} + lim_{x \to 2^{-}} a x - lim_{x \to 2^{-}} 3 a$

Step 1.3

Move the term

4

$4$ outside of the limit because it is constant with respect to

x

$x$ .

4 lim_{x \to 2^{-}} e^{x - 2} + lim_{x \to 2^{-}} a x - lim_{x \to 2^{-}} 3 a

$4 lim_{x \to 2^{-}} e^{x - 2} + lim_{x \to 2^{-}} a x - lim_{x \to 2^{-}} 3 a$

Step 1.4

Move the limit into the exponent.

4 e^{lim_{x \to 2^{-}} x - 2} + lim_{x \to 2^{-}} a x - lim_{x \to 2^{-}} 3 a

$4 e^{lim_{x \to 2^{-}} x - 2} + lim_{x \to 2^{-}} a x - lim_{x \to 2^{-}} 3 a$

Step 1.5

Split the limit using the Sum of Limits Rule on the limit as

x

$x$ approaches

2

$2$ .

4 e^{lim_{x \to 2^{-}} x - lim_{x \to 2^{-}} 2} + lim_{x \to 2^{-}} a x - lim_{x \to 2^{-}} 3 a

$4 e^{lim_{x \to 2^{-}} x - lim_{x \to 2^{-}} 2} + lim_{x \to 2^{-}} a x - lim_{x \to 2^{-}} 3 a$

Step 1.6

Evaluate the limit of

2

$2$ which is constant as

x

$x$ approaches

2

$2$ .

4 e^{lim_{x \to 2^{-}} x - 1 \cdot 2} + lim_{x \to 2^{-}} a x - lim_{x \to 2^{-}} 3 a

$4 e^{lim_{x \to 2^{-}} x - 1 \cdot 2} + lim_{x \to 2^{-}} a x - lim_{x \to 2^{-}} 3 a$

Step 1.7

Move the term

a

$a$ outside of the limit because it is constant with respect to

x

$x$ .

4 e^{lim_{x \to 2^{-}} x - 1 \cdot 2} + a lim_{x \to 2^{-}} x - lim_{x \to 2^{-}} 3 a

$4 e^{lim_{x \to 2^{-}} x - 1 \cdot 2} + a lim_{x \to 2^{-}} x - lim_{x \to 2^{-}} 3 a$

Step 1.8

Evaluate the limit of

3 a

$3 a$ which is constant as

x

$x$ approaches

2

$2$ .

4 e^{lim_{x \to 2^{-}} x - 1 \cdot 2} + a lim_{x \to 2^{-}} x - 3 a

$4 e^{lim_{x \to 2^{-}} x - 1 \cdot 2} + a lim_{x \to 2^{-}} x - 3 a$

Step 1.9

Evaluate the limits by plugging in

2

$2$ for all occurrences of

x

$x$ .

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Step 1.9.1

Evaluate the limit of

x

$x$ by plugging in

2

$2$ for

x

$x$ .

4 e^{2 - 1 \cdot 2} + a lim_{x \to 2^{-}} x - 3 a

$4 e^{2 - 1 \cdot 2} + a lim_{x \to 2^{-}} x - 3 a$

Step 1.9.2

Evaluate the limit of

x

$x$ by plugging in

2

$2$ for

x

$x$ .

4 e^{2 - 1 \cdot 2} + a \cdot 2 - 3 a

$4 e^{2 - 1 \cdot 2} + a \cdot 2 - 3 a$

4 e^{2 - 1 \cdot 2} + a \cdot 2 - 3 a

$4 e^{2 - 1 \cdot 2} + a \cdot 2 - 3 a$

Step 1.10

Simplify the answer.

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Step 1.10.1

Simplify each term.

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Step 1.10.1.1

Multiply

- 1

$- 1$ by

2

$2$ .

4 e^{2 - 2} + a \cdot 2 - 3 a

$4 e^{2 - 2} + a \cdot 2 - 3 a$

Step 1.10.1.2

Subtract

2

$2$ from

2

$2$ .

4 e^{0} + a \cdot 2 - 3 a

$4 e^{0} + a \cdot 2 - 3 a$

Step 1.10.1.3

Anything raised to

0

$0$ is

1

$1$ .

4 \cdot 1 + a \cdot 2 - 3 a

$4 \cdot 1 + a \cdot 2 - 3 a$

Step 1.10.1.4

Multiply

4

$4$ by

1

$1$ .

4 + a \cdot 2 - 3 a

$4 + a \cdot 2 - 3 a$

Step 1.10.1.5

Move

2

$2$ to the left of

a

$a$ .

4 + 2 a - 3 a

$4 + 2 a - 3 a$

4 + 2 a - 3 a

$4 + 2 a - 3 a$

Step 1.10.2

Subtract

3 a

$3 a$ from

2 a

$2 a$ .

4 - a

$4 - a$

4 - a

$4 - a$

4 - a

$4 - a$

Step 2

Evaluate

x^{3} + a x^{2} + 5

$x^{3} + a x^{2} + 5$ at

2

$2$ .

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Step 2.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

{(2)}^{3} + a {(2)}^{2} + 5

${(2)}^{3} + a {(2)}^{2} + 5$

Step 2.2

Evaluate.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Raise

2

$2$ to the power of

3

$3$ .

8 + a {(2)}^{2} + 5

$8 + a {(2)}^{2} + 5$

Step 2.2.1.2

Raise

2

$2$ to the power of

2

$2$ .

8 + a \cdot 4 + 5

$8 + a \cdot 4 + 5$

Step 2.2.1.3

Move

4

$4$ to the left of

a

$a$ .

8 + 4 a + 5

$8 + 4 a + 5$

8 + 4 a + 5

$8 + 4 a + 5$

Step 2.2.2

Add

8

$8$ and

5

$5$ .

4 a + 13

$4 a + 13$

4 a + 13

$4 a + 13$

4 a + 13

$4 a + 13$

Step 3

Since the limit of

4 e^{x - 2} + a x - 3 a

$4 e^{x - 2} + a x - 3 a$ as

x

$x$ approaches

2

$2$ from the left is not equal to the function value at

x = 2

$x = 2$ , the function is not continuous at

x = 2

$x = 2$ .

Not continuous

Step 4