Calculus Examples

$lim_{x \to π} {(x - π)}^{2} \cos (\frac{1}{x - π}) + 3 x^{2}$

Step 1

Evaluate the limit.

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Step 1.1

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

$lim_{x \to π} {(x - π)}^{2} \cos (\frac{1}{x - π}) + lim_{x \to π} 3 x^{2}$

Step 1.2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$lim_{x \to π} {(x - π)}^{2} \cos (\frac{1}{x - π}) + 3 lim_{x \to π} x^{2}$

Step 1.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$lim_{x \to π} {(x - π)}^{2} \cos (\frac{1}{x - π}) + 3 {(lim_{x \to π} x)}^{2}$

Step 2

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

$lim_{x \to π} {(x - π)}^{2} \cos (\frac{1}{x - π}) + 3 π^{2}$

Step 3

Consider the left sided limit.

$lim_{x \to π^{-}} {(x - π)}^{2} \cos (\frac{1}{x - π})$

Step 4

Make a table to show the behavior of the function ${(x - π)}^{2} \cos (\frac{1}{x - π})$ as $x$ approaches $π$ from the left.

$\begin{array}{c} x & {(x - π)}^{2} \cos (\frac{1}{x - π}) \\ 3.04159265 & - 0.00839071 \\ 3.13159265 & 0.00008623 \\ 3.14059265 & - 0.00000011 \end{array}$

Step 5

As the $x$ values approach $π$ , the function values approach $0$ . Thus, the limit of ${(x - π)}^{2} \cos (\frac{1}{x - π})$ as $x$ approaches $π$ from the left is $0$ .

$0$

Step 6

Consider the right sided limit.

$lim_{x \to π^{+}} {(x - π)}^{2} \cos (\frac{1}{x - π})$

Step 7

Make a table to show the behavior of the function ${(x - π)}^{2} \cos (\frac{1}{x - π})$ as $x$ approaches $π$ from the right.

$\begin{array}{c} x & {(x - π)}^{2} \cos (\frac{1}{x - π}) \\ 3.24159265 & - 0.00839071 \\ 3.15159265 & 0.00008623 \\ 3.14259265 & - 0.00000092 \end{array}$

Step 8

As the $x$ values approach $π$ , the function values approach $0$ . Thus, the limit of ${(x - π)}^{2} \cos (\frac{1}{x - π})$ as $x$ approaches $π$ from the right is $0$ .

$0 + 3 π^{2}$

Step 9

Add $0$ and $3 π^{2}$ .

$3 π^{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$3 π^{2}$

Decimal Form:

$29.60881320 \dots$