Calculus Examples

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

Step 1

Find the limit of $4 e^{x - 2} + a x - 3 a$ as $x$ approaches $2$ from the left.

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Step 1.1

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

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

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 1.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.4

Move the limit into the exponent.

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

Step 1.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 1.6

Evaluate the limit of $2$ which is constant as $x$ approaches $2$ .

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

Step 1.7

Move the term $a$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.8

Evaluate the limit of $3 a$ which is constant as $x$ approaches $2$ .

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

Step 1.9

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 1.9.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

Step 1.9.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

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 $- 1$ by $2$ .

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

Step 1.10.1.2

Subtract $2$ from $2$ .

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

Step 1.10.1.3

Anything raised to $0$ is $1$ .

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

Step 1.10.1.4

Multiply $4$ by $1$ .

$4 + a \cdot 2 - 3 a$

Step 1.10.1.5

Move $2$ to the left of $a$ .

$4 + 2 a - 3 a$

Step 1.10.2

Subtract $3 a$ from $2 a$ .

$4 - a$

Step 2

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

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Step 2.1

Replace the variable $x$ with $2$ in the expression.

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

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 $2$ to the power of $3$ .

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

Step 2.2.1.2

Raise $2$ to the power of $2$ .

$8 + a \cdot 4 + 5$

Step 2.2.1.3

Move $4$ to the left of $a$ .

$8 + 4 a + 5$

Step 2.2.2

Add $8$ and $5$ .

$4 a + 13$

Step 3

Since the limit of $4 e^{x - 2} + a x - 3 a$ as $x$ approaches $2$ from the left is not equal to the function value at $x = 2$ , the function is not continuous at $x = 2$ .

Not continuous

Step 4